Using America to refer to every country in North and South America is imprecise at best, if not outright disingenuous. Nobody thinks you're talking about Brazilians or Argentines or Canadian when you say "American."
That's us. It's our word. You can pry it from our cold dead lips.
Your America-first approach to naming is fine in American circles but don't be surprised when other people use other terminology. No, America is indeed a continent. I have a lot of Brazilian/Peruvian/Canadians colleagues and have been in contexts where we've discussed American (continent) vs European points of views.
I am not sorry if my attempt to use precise language when precision is needed is "disingenuous" to you. What a weird thing to get rude about.
That's us. It's our word. You can pry it from our cold dead lips.