US and USA are not adjectives: they are nouns that can potentially be used attributively, with the same meaning as the corresponding adjectives (if such exist). Headlinese has slightly different rules, but in normal, non-Headlinese language use:
USA is not used attributively because it is an abbreviation of a compound country name (United States of America), and these cannot be used attributively. The corresponding adjective is American.
US is used attributively because it is an abbreviation of a simple, non-compound country name (United States), and these (rare though they are) can be used attributively. There is no corresponding adjective.
Country names: noun adjuncts and true adjectives
Generally speaking, most country names in English can be used adjectivally, as noun adjuncts; that is, they can be used with the force of an adjective (meaning ‘of Country’ or ‘from Country’) to modify a noun (“the X noun”). Like all nouns, however, they cannot be used as predicative adjectives (as in “the noun is X”):
[noun] The Zimbabwe economy / *He is Zimbabwe
(Compare attributive “a golf shoe”, but predicative “*the shoe is golf”.)
A lot of country names have corresponding true adjectives that are used instead of the country name when adjectival use is required. These can be used both attributively and predicatively:
[adjective] The Zimbabwean economy / He is Zimbabwean
In most cases, blocking prevents both the adjective and the noun from being in common use at the same time for (attributive) adjectival use: if there is a true adjective, that is normally preferred over the country name. The exception is in Headlinese, where whichever form is shorter is often used. Even in Headlinese, however, noun forms are never used predicatively:
[language use] Attributive / Predicative
[normal] A Canadian city / He is Canadian (adjective)
[normal] *A Canada city / *He is Canada (noun)
[Headlinese] Canadian MP to stand down / The MP is Canadian (adjective)
[Headlinese] Canada MP to stand down / *The MP is Canada (noun)
Country names: simple and compound
This all applies to what I shall—for lack of a better term—call simple country names, that is, names that consist only of one or more constituents that denote territories in some way. (Constituents being here more or less equivalent to noun phrases—so [South Africa] is one constituent, but [São Tomé] and [Príncipe] is two, combined with the coordinator ‘and’.)
Simple country names contrast with what I shall call compound country names, which are names that can be split up into a simple country name and a constituent that is not a country name. Usually, this non-name constituent denotes a type of governance (kingdom, republic, federation, etc.), and compound names normally have the structure [governance type] of [simple country name] (combined by the subordinator ‘of’):
[United States] of [America]
[United Kingdom] of [Great Britain and Northern Ireland]
[Federation] of [Micronesia]
[People’s Republic] of [China]
Unlike simple country names, compound country names cannot be used adjectivally, even in Headlinese. The only option available to turn a compound name to adjectival use is to break the compound up into its constituents and use the appropriate (noun/adjective) form of the constituent that is a simple country name (syntactically the head of the noun phrase):
Former Yugoslav Republic of Macedonia -> [noun] Macedonia / [adjective] Macedonian
People’s Republic of China -> [noun] China / [adjective] Chinese
United States of America -> [noun] ?America1 / [adjective] American
We cannot say “a People’s Republic of China city” or even in Headlinese “People’s Republic of China MP to stand down”; but we can say “a Chinese city” and “Chinese MP to stand down”, and in Headlinese, we can say “China MP to stand down”.
In normal language usage, it is exceedingly common to break up compound names and just use the embedded simple name to refer to the country: we talk about Macedonia, Micronesia, and China much more commonly than about the Federation of Micronesia, the Former Yugoslav Republic of Macedonia, or the People’s Republic of China.
Some of these compound names have established abbreviations (like FYROM and PRC for the two examples above), since the full versions tend to be rather long.2 These abbreviations normally function in the same way as simple country names: that is, they can be used adjectivally (though only attributively), but only in Headlinese:
[normal] A Macedonian town / *A FYROM town
[Headlinese] FYROM MP to stand down / Macedonian MP to stand down
This in particular is the reason why USA is not used attributively in normal language use. I’m not entirely convinced that it is completely unused in Headlinese (at least outside the States), but even in Headlinese, it is very rare. This is because the US is one of a small set of special cases.
Special cases: US and UK
There are a few special cases: countries that have compound names and where English is the main (if not only) commonly spoken language. We tend to talk about our own countries a lot more than we do about other countries, and while we may be all right talking about a seven-syllable São Tomé and Príncipe however often the name pops up in conversation, constantly talking about a massive fifteen-syllable United Kingdom of Great Britatin and Northern Ireland is far too much of a mouthful for us lazy humans. So we shorten the names.
In some cases, we can do this just by using the territory name alone (the Republic of Ireland is frequently referred to as just Ireland, for example), but not always. In some cases, this leads to ambiguity: Ireland is also the name of the entire island (encompassing Northern Ireland as well), and America is also the entire continent. In other cases, the territory name is itself rather a mouthful and not easily simplified: Great Britain and Northern Ireland is still nine syllables.
In these cases, the alternative solution is to ditch the territory name and use the ‘governance type’ constituent on its own. Since this is done mainly by people who actually live in the country, it’s usually obvious which republic, federation, kingdom, or other state type they’re talking about—and luckily, the three examples mentioned above are the only common ones, and they have three different governance types, so you won’t get them mixed up:
the Republic [of Ireland]
the United States [of America]
the United Kingdom [of Great Britain and Northern Ireland]
The resulting shortened names are, as you can see, not compound anymore. They contain only a single constituent, which (although it’s a governance type) now refers synecdochically to the territory. So it’s now a simple country name, rather than a compound country name. And to no one’s surprise, these new simple names function just like all other simple country names: they can be used adjectivally, though only attributively. Since the shortened names have no corresponding adjectives (the normal adjectives being correspondent to the full, compound names), there is no blocking process to favour an adjective over the noun, and the attributive forms are used even outside Headlinese, as in the example in the question:2
He’s a United States senator
Even the shortened names the United States and the United Kingdom are still five and six syllables long, respectively, and it’s not entirely surprising that they should be abbreviated in the same way that many compound names (including the United States of America) are. So United States becomes US, and United Kingdom becomes UK. Republic is short enough—no need to further abbreviate that.
These are in the somewhat unusual position of being abbreviations of simple nouns rather than compound nouns, and the limitations for using them adjectivally are a bit different as well. Compound noun abbreviations are used in the same way as simple country names: attributive use only, and only in Headlinese. Attributive use is not found in normal use, where instead you use the preferred attributive form of the head noun in the non-abbreviated compound name; i.e., you basically undo the abbreviation before applying the normal compound-name rules (leaving you with a nice and short form). Here, though, undoing the abbreviation leaves you with rather a long simple noun, and with no other way of shortening it.
This discrepancy between UK/US and other abbreviations is apparently motivation enough to allow normal, non-Headlinese attributive use of the abbreviations. The only alternative would be to use the non-abbreviated forms, and five syllables for your own country is apparently just too much to deal with.
Of course, this means we have two super-practical (disyllabic!) ways of referring to ‘our own’ countries, which is just what we want. Naturally, their usage is extremely widespread. I suspect the common use of US and its immediate similarity to USA is a big part of the reason why even Headlinese prefers the former to the latter. It’s basically just blocking again.
1 This form is somewhat questionable. Theoretically, “America MP to stand down” works in Headlinese; but since the commonly used abbreviation (see further down) is so much shorter, Headlinese, which goes for shorter forms, never favours it.
2 It is interesting that the United Kingdom of Great Britain and Northern Ireland does not have an abbreviation at all. The only abbreviation found is UK, which is from the shortened form (see “Special Cases”).
3 Republic is a bit of an exeption to this: it is rarely used attributively outside Headlinese. When talking about the Republic of Ireland and Northern Ireland in opposition to one another, Irish people in general avoid adjectival use entirely and recast their sentences to say “in the Republic” and “in the North/Six Counties”, respectively.