Traditionally, due to the technical limitations of printing, capitalized letters were not written with an accent (note this does not change the pronunciation), and you'll see unaccented capitals in plenty of older books and journals, at least.
At present, there are no longer technical limitations preventing the typing and printing of capitals without accents, so I'd suggest including the accents.
The calque of America is "Amérique." I suspect Amérique in most francophone countries defaults as a reference to all of North and South America.
If it does not, I can say with confidence that I seldom heard the locals in France refer to the US as anything but "les États-Unis" and "les States."
If you look out for example Here, you get something like this:
America is the name of a whole continent. United States of America means that the United States belongs to America and NOT that America belongs to the United States. So, next time you want to refer to The United States of America, you can do it as U.S. or the States or whatever you want but not as only America.
What you say is true only in the most pedantic sense. If someone said "I'm going to America," no one would ever think "oh, he's going to American the continent." or ask "Oh do you mean North or South America?" - they would assume American equated to the United States. Yes, it could be perceived as arrogant or colonizing, but it's also a reality that this linguistic understanding between the two terms exists. America should be accepted.