Actually, no, we do use alot of everyday and semi-regular use words from those languages, the below is an example of 10 words each from from Greek and latin that are used regularly or semi-regularly.
Greek:
Anonymous, Angel, Bacteria, Baptism, Geography, Genetic, Encyclopedia, Energy, Helicopter and History
Latin:
Animal, Artificial, Cave, Curious, Family, Feral, Liberal, Local, Veteran and Victory.
Those are a tiny proportion of everyday English vocabulary, and several are very specific. More importantly, they are all nouns and adjectives.
The actual basic components of the English syntax and grammar are nearly all Old English, and those that are not are generally Old Norse.
Let me give you some examples, 'a', 'an', 'the', 'them', 'they', 'you', 'he', 'she', 'on', 'be', 'to', 'of', 'have', 'I', 'not', 'with', 'do', 'but', 'from', 'we', 'say', 'would', 'what', 'why', 'or', 'when'. Need I go on?
The issue is not that there are no commonly used Latinate or Romance words in English, it's that speaking from the perspective of syntax there is nothing remotely resembling Latin cognates or structure in English sentences. There is a vast difference between two languages sharing similar or identical syntax, and another between languages that happen to share some nouns and technical terms. English is linguistically speaking firmly in the Germanic camp.