Neither US nor USA is the name of the country in question. It is an acronym. One does not extend acronyms. Perhaps you could come up with even a single example to the contrary? Would you call someone from the UK a UKian or someone from the former USSR a USSRian.
If you are going to add an extension you have to use the actual name which is technically The United States of America or at least The United States or United States for short. The fact that the US is a two word country and one that ends in an 's' makes this rather awkward unfortunately.
Since the full name of the country is 'The United States of America". You could properly refer to someone from that country as an "United States of American" I suppose. That would be the standard rule to apply for most countries that end in 'a'. Just add an 'n'. Since there is an easy rule to apply there isn't much point in talking further about usage of the complete name. Probably the only reason not to follow that simple rule is laziness. If you object to 'American' as being imprecise then just use 'United States of American' instead.
If you think United States of American is too long then you could try using just United States as the name and try to apply one of the usual rules to that as a country name. United States is one of the relatively few two word countries so I'll use examples from that domain.
Sri Lanka ends in an 'a' so the result is obvious. Just add an 'n'. The same rule can be applied to Costa Rica, Saudi Arabia and South/North Korea.
To New Zealand an -er is added probably because it ends in 'land' which is similar to island and -er is usually applied in such cases. United States would become United Stateser.
The Philippines -> Philippine. In that case you just drop the 's'. So someone from the United States would be called a United State.
Belarus -> Balarusian. There's an example of your prefered 'ian' ending. In that case you would call someone from the US a United Stateian or maybe you could drop the 'e' resulting in United Statian.
Cyprus -> Cypriot. So United Statiot I suppose if you drop the 'e'.
Honduras -> Honduran. The 's' is dropped and then the usual rule is applied to countries ending in 'a'. Just add an 'n'. So to follow this rule United States would become United Staten if you ignore the fact that you have an 'e' instead of an 'a' at the end.
If you follow the rule for countries that end in 'e' then you might follow the rule applied to Chile and just add an 'an'. So you'd have United Statean. I kind of like that one.