Canadians and Greenlanders don't refer to themselves as North Americans but no one is stopping them, certainly not Americans.
Argentinians and Chileans don't refer to themselves as South Americans..
Mexicans and Panamanians don't refer to themselves as Central Americans.
I think it's safe to say that the real reason no one does this is because it's unnecessarily performative. If someone asked me or a northern neighbor where were from and we both said "North America" then their next obvious question is, "Which country?" Since we all know that's where things are going to wind up, why not start there in the first place and save the unnecessary drama?
One of the things about living in or being an American is that a lot of people from all over the world come here. A lot.
Never in my life have I asked someone where they are from and they've chosen to say their continent over their country. It just doesn't happen. They say Ireland, Ivory Coast, El Salvador, Spain, Uraguay, Japan, Laos, Zimbabwe, Antigua, etc.
It's interesting that you're demanding Americans set themselves apart from how everyone else in the world does things because you've bought into a ridiculous idea that Americans think they're the only Americans.
Shortening a long country name when referring to yourself isn't unique to Americans either. If you ask someone from The Democratic Republic of Congo where they're from they'll say either "the Congo" or DCR. We all know what they mean. It's not their fault their country has such a long name; they're not beholden to tie their tongue in knots or deal with cumbersome linguistics just to make things more palatable for you.
Some folks still refer to themselves as what they were historically rather than use a long country name. I've never heard someone announce themselves as being from The United Kingdom of Great Britain and Northern Ireland; they'll say they're from England, Scotland, or Wales though. Sometimes they'll say they're from the UK, just as Americans are equally as likely to say they're from the US. Depends on sentence structure and cultural area of the US the person is from. Nobody goes on an apoplectic tirade over people from the UK not saying that whole long spiel.
More in line with what Americans do... Nobody says they're a Former Yugoslav Republic of Macedonian. That sounds stupid, so they just say they're Macedonian because it's linguistically easier and everybody knows what they mean. No one would make such a ludicrous suggestion that Macedonians believe they are (were) the only Yugoslavians or the only Eastern Europeans.
If you look around the world, people from the US calling themselves American for convenience and maintaining speech flow isn't any different than what others around the world with the same problem do. It's exactly the same.
