There's not a clear and conscience alternative to "American." If you're trying to differentiate us from other people from the Americas you'd just say US Citizen. And while yes the entirety of this hemisphere is some variation of America be it North, Central, or South the other countries here have distinct names and we really don't.
At the risk of sounding like a typical US asshole, here goes nothing. This is how I've explained it to friends from Europe and it seemed to help.
If Brazil had decided to go by the name "United States of Brazil" we would still call them Brazilians because there is another country with the title "United States" that also exists. Similar to how we call people from the Peoples Republic of China, Chinese. We don't call them "People's Republicans" because that's a title not a unique identifier or name. What if that same country decided to go by the name the Peoples Republic of Asia instead, would we call them Peoples Republicans or would we call them Asians?
The title "United States" is telling you that this area is united together and the borders represent states, not country's. "America" tells you where those united states are, the continent of America. The term "American" is generalized and honestly doesn't accurately represent the vast cultural differences within the United States. The states often have their own rights and laws separate from the US government and also unique cultures. Ideally we would be called by our states name for its citizens like Californian or New Yorker, for example. Similar to how you would refer to people from Europe as European unless you wanted to be specific to Italy, then you'd say Italian. But sometimes you need a general term, hence "American."
All that being said, it is problematic and a massive reminder of this country's bloodthirsty and genocidal colonization of a large part of North America. Looking at the country's past shows that they were very much trying to also get central and south America as part of the United States. What better way to propagandize and make it look like they had every right to the rest of the Americas than to make it appear as though this country or that country already was America and therefore should be part of these United States? But however problematic it is this is the name we have now, for better or worse.
As an addendum of sorts. We Indigenous Americans would often much rather be called by the names of our sovereign Nations yet everyone calls us Native American. Why is that? Food for thought that might help with understanding the problematic struggle we have here. It's not simply us as citizens that perpetuate the issue, it's a global colonization effort whether the others realize they are participating or not. (Spoiler: they realize)
tldr: because colonization + United States is a title not a name