Some observations/suggestions from my own experience for you to take or leave.
I wasted a lot of time (and research) until I created a specific definition of "better outcome" for my timeline. "Better performance" can be measured in many ways over short, intermediate and long time periods. Tactical performance? Strategic objectives? Shortening the war? Fewer casualties?
IMHO a lot of time and discussion is wasted on "which gadget is best" (take your pick of weapon type) for things that did not have a large impact on the outcome or timeline for WWII. For example, US drops the BAR and adopts the Bren Gun in 1939. So how much does that shorten the war? How important is a better RPG/bazooka/LAW...? Extremely important to the grunt trying to stop an enemy tank or silence a bunker, but personally I would rather have a tank or three.
During WWII, the number one killer and cause of non-fatal wounds in allied soldiers is ARTILLERY (including mortars). Period.
I believe the biggest change is ground force performance is combined arms warfare (and its myriad subcomponents). Many evolutions and call it what you may. In 1939, the Germans are on the neonatal side of it, but they move the bulk of their guns and supply with horses (and many other immature concepts) and the bulk of their army is foot mobile beyond the nearest railhead. That is a doctrine battle. Everyone has some idea of the concept, but none (including the Germans) have gone far enough. All the fire control arguments (e.g., British versus American) are not nearly as important as using tanks and air support to better effect (and training). Decentralization is very important and inter-branch conflicts (which is a money battle at a fundamental level). Push the tanks, tank destroyers (if you keep them) and heavy weapons forward and train in their use at the company level. Train tank-infantry teams and improve coordination.
Both the British and Americans need to throw serious cold water on strategic bombing. Again, serious doctrine fight on both sides of the Atlantic. Better bombers only make it worse in my opinion. Bombers consume workers, crew, aluminum, steel, engines and money well in excess to their value. NEITHER SIDE can hit the broad side of a barn with technology available in the next several decades. The need for large numbers of aircraft (even if they adopt a more rational doctrine) results in numerous trans-Atlantic problems. Need puts a lot of trash in production, but the primary limiting factor on both sides of the Atlantic is engines.
From a technology aspect, solve the HORSEPOWER-TO-WEIGHT (HTW) problem everyone faces. That will shorten the war, but requires a serious infusion of money a lot earlier. I love the discussions about "get the XXX wonder fighter into production in 194x". Powered with what? The Merlin engine (not the 1940 version) solves the high altitude interceptor problem, but not the ground attack fighter problem.
HTW (weight and size being a trade-off) is also the limiting factor on the tank (and truck) side of the equation. The push to build quantity rapidly creates a lot of mistakes. So, my two cents on the Sherman tank - what should one switch to in late 1941 when you are lining up production for 1942? One needs a lot of post-combat understanding to build a better tank in 1942. Remember, at this time the British are "up-gunning" to the 6pdr and German tanks with a short-barreled 75 are appearing in the desert.
Centimetric (and X band) radar is an obvious technological need. Don't forget counterbattery, even with teething problems it beats sound. Reality hits with a multitude of needs.
Improved ASW doctrine needs a lot of help if you want it early. Many years of lessons to be learned. When does the US start building escorts and what precipitates a change that puts ASW on par with the need for battleships, cruisers, fleet destroyers and carriers? How do the allies come to realize that sonar/ASDIC has some significant limitation. What precipitates the early introduction of escort carriers for ASW patrol? The British aren't there until 1941 and their early attempts met with disaster. The move to convoys without adequate escorts was met with higher losses. The RN's hard look at real analytics did not happen until 1943. Should the Americans have sent their crews through the British ASW training? Absolutely, but the British barely had the capacity to train their own.
Oh, and I must comment on the idea of "better" US escorts. Bethlehem Hingham fulfilled a British order for 12 Buckley class DEs in late '43, laying the first keel on 15 September and delivering the last ship on 8 February (total build times ranging from 96 to 121 days). Granted, not their first rodeo for this ship type, but escorts were needed in big numbers. Hard to improve on "good enough" built quickly. In all fairness, the first two DEs build in their yard took 283 and 324 days, respectively but they were at ~180 days or less by their 12th build (2Q43).
On the production side, the US (and British) created significant unexpected consequences by planning on a larger ground force than needed. If you peel several layers off the US onion concerning decisions made about weapons, ponder the impact of believing you need to equip 200 divisions and transport them across the ocean. Oh f*ck! Now, pile on the stated desire to conduct a cross-channel attack sometime between 3Q42 and 2Q43. Stuff needs to get made most Rikki-tic and Army Ground Forces needs to make tough decisions in a hurry (with the currently available doctrine). Replace Lesley McNair any time you want and the next poor bastard is pretty much under the same constraints. Might make one or two better decisions, but he will be forced to rapidly build a large Army with what he can put into mass production in 1942. And, he will be forced to consider every cubic foot and short ton of cargo ship space.
No argument from me about canning MacArthur, but one needs a massive "go fund me" to stand-up and train the Philippine Army in early to mid 1940 or send US troops who are not available (or able to be adequately supplied half-way around the world). Problems always meet with financial reality and conflicting requirements.
Concerning an early butt kicking. I have seriously pondered whether this is necessary. Name a western military organization that would actually adopt someone the equipment and doctrine of an ally in the 1930s. Don't hold your breath. Find a democratic government that would agree to pay royalties on equipment rather than fund their own designs. Now, go on to examine WWII in practice. Virtually all significant changes accelerate in the face of tough lessons. Arguments for A versus B are overly subject to opinion until the selection does not perform. Furthermore, consistent with human nature bad decisions need to result in significant failure before change becomes imperative. Improvements get labeled as incremental, etc. This is a human, not American problem. Sadly, there is no substitute for direct combat experience. None. Advice, observation, WWI experience are nearly universally of minimal value in preparation for WWII (again, applicable to all the allied powers).
I personally believe the Japanese suffered significantly later in the war because they didn't get their butt kicking until it was too late. They were lucky beyond rational expectation in early 1942, feeding all their faulty concepts. Any failure was simply attributed to the field commander and their real issues were not recognized until it was way too late to help.
So, a loss like Kasserine Pass opens eyes in a hurry. Was the prior lack of accepting British advice purely stupid Anglophobia? That is obviously a factor, but put yourself in Marshall's shoes in 1941. Your current doctrine and training methods conflict with the advice you are receiving from the British. Most of the fatherly advice was accompanied by the suggestion that the British be put in command. One can argue that the US response was Anglophobic; however examine the situation to date. RAF Fighter Command aside, how well have the British performed in the field. Is that not the ultimate test of weapons and doctrine? France, North Africa (after the Germans arrive), Greece and Crete is not a strong endorsement. While they have learned a lot of hard lessons, the evidence that they have reached the proper conclusion is still lacking. Noting that reaching the proper conclusion is the key to gaining value from observation of prior battles. Everyone views circumstances through a lens filtered by their own beliefs. Just a human thing. (Note that the IJA thinks they are shit hot)
In addition to doctrine changes, the early war allied problems are in part related to the whole concept of a run-up with time to "mobilize" before fighting. While this worked (somewhat) in WWI, it did not work at all in WWII for any allied nation. When I speak of "mobilization" I am referring to the centuries old practice of having a small peacetime Army and then quickly fill divisions with reservists and draftees who have a minimum of training (as well as outdated equipment). This creates a multitude of problems and was clearly obsolete (see Battle of France).
Another addition to the problem list is convincing civilian manufacturers to dedicate space, people and dollars to war production. Very hard to do during OTL. Government money only solves part of the problem as reallocation of square footage and personnel is risky business relative to the whims of Congress. Expansion is also a risk to the existing businesses, so someone needs to come up with the idea of the government-owned, civilian-operated plant for aircraft and AFV. Otherwise the government gets in the aircraft and transportation business which would result in a radical shift of campaign funds to the other party.
Last problem is R&D. Imagination is fairly easy to solve by creating an innovator, but getting practical specifications and avoiding capability creep is always an issue. One needs to innovate and go beyond what many will believe is feasible. IMHO that requires someone to take the financial risk to produce the innovation and then sell it to the military.
Some potential solutions I have imagined. The reader must judge the quantity of alcohol involved in these conclusions.
From a purely American perspective, two changes could potentially have a positive impact on training. This assumes improvement in doctrine and the availability of those fiddly bits like rifles, boots, housing, ammunition, etc. BTW, infantry doctrine was already quite similar to Britain. First, allocate draftees (and recruits to the extent possible) to branches and schools after a common basic training. During OTL branch/school assignments were made at induction centers, even for draftees. Add some infantry training (including field exercises) to all members post boot camp (like the USMC). During OTL, troops were trained (from basic on) in either a branch school or infantry division. The first issue (allocation at induction) created a real problem with talent (physical and mental) distribution. The second resulted in all types of disparities in training methods and quality.
An ABSOLUTE MUST (along with the change in training methods) is to build training facilities to handle the load. Too much "training time" was spent on infrastructural and cadre issues during OTL. Lots of troops sitting around getting bored during 1940-1941.
The War Department MUST be willing to stand-up fewer divisions until the pipeline gets rolling. This requires a recognition that they can achieve better combat power with proper application of better doctrine.
To have any hope of moving away from the "mobilization" mode of action, the US needs to agree to having a much larger standing Army in the mid 1930s. Good luck with that, however fixing the training program will help. One could potentially argue that the US relied too much on large scale maneuvers and not enough on small unit tactics and individual skills (again, universal Western issue factored to local scale).
Improved engines need an incentive (i.e., money) as well as knowledge that 1,000hp is inadequate for combat aircraft. Perhaps prize money for air races? Tractor pulls? Probably crazy, but how does one get to the need for a 2,000hp radial engine?
My prior research into the subject of engines for tanks indicates that ~750hp is the maximum for a naturally aspirated (no ECM) diesel that will fit in an armored vehicle. Obviously there are diesels that fit in submarines and locomotives that are more powerful. This is a massive achievement for the time. Does anyone even believe it is feasible? PS, don't forget the "bridge problem" - 40t is a serious challenge for bridges in Europe (most everywhere for that matter). It is also an issue for assault bridging and ferry operations of the OTL variety.
Testing money from Congress and somehow enforce outside objective review by someone without a vested interest. Very difficult to accomplish in a representative democracy of free market capitalists (or any other ilk for that matter). This could solve a myriad of problems.
Another BSC idea is for the US to have a war in the mid-1930s to gain the experience (i.e., failures) necessary to drive improvements. I am unable to imagine where or against whom.
I personally like the "gun porn" on these forums as much as anyone else, but a better LMG or switching to an intermediate round at an earlier date really doesn't change that much. Definitely better decisions, but I'd rather have ground-breaking doctrine and well-trained troops with a BAR. Want to help the grunt? Improve his web gear and lighten his frigging load! I would mention better rations, but that would strain even a powerful ASB (historically speaking anyway).
Hope this musing helps a little. Maybe reaffirms what you have already considered.