That’s a gross simplification.
First, much of said “fucking it up” was done by Norse influence due to all the Danes. Started a bit earlier than 1066, something-something Canute, something-something Danelaw, something-something great pagan army.
Second, Norman French was one of two clearly separable waves of “French” influence, and the commoners’ Germanic language after it was still very different from what happened after the other wave.
Third, that other wave was Angevin French, which, yeah, made a big influence, in huge part because it happened in the time when English kings were trying to conquer much of France for themselves, and didn’t see themselves as English kings really.
It has a soft flavor. I don’t put it into anything spicy, and probably won’t be noticeable with the way Americans seem to do seasoning. But if I’m making a soup with some meat and potatoes and various vegetables, I’ll put it in, it’ll be noticeable.
If you just boil beef with and without it, you’ll feel the difference the most, I think.