It's sort of how USians say "zip codes" for foreign addresses when they mean "postcodes" (the Zone Improvement Plan, from which ZIP is taken, is a US thing).
"Citizen of the USA" is a mouthful. People tend to use "American" but that's ignoring the fact that there are 35 countries in America.
A lot of languages have a word like USian, e.g. "états-unien" (French), "estadounidense" (Spanish). I have used "USAmerican" myself when "American" would be ambiguous.
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.
If there were another North or South American country that went by America (e.g. if Brazil was instead called the "Eastern American Republic") then what you say would make sense. As is, it just sounds like you're trying to stick it to the man.