Sure, but many of those words for specialised doctors came to English through French, not directly from Latin or Greek. And I don’t think that you can reasonably argue that English words with French origins aren’t by now a native part of the language. We use many of the same names in Dutch too, coming from French loanwords.
French remained influential in the courts, higher education, and elite society long after it stopped being the “official” language. That last part is totally right.