I tried but failed to wrap my head around the grammar for the term "American-born" (and all other similar expressions - "Canadian-born", "French-born", ...etc). Is it a set phrase or is there some grammar rule why the term has to be American-born, not America-born ?
There are 2 ways to express the country of birth.
born as a(n) [American, Canadian...]
born in [America, Canada...]
It is similar with the pattern:
(adjective) - ("past participle" of a noun)
short-sleeved = with a short sleeve
fast-paced = with a fast pace
(adjective) - (past participle)
American-born = born in America
"American-born" means that someone was an American citizen from birth. It doesn't necessarily mean they were physically "Born in the USA."
"America-born" does mean they were physically born in the USA. For most countries, that does not necessarily mean they are a citizen of the country where they were born, but the USA is an exception to that general rule, because of the 14th Amendment to the Constitution.
Note: there are a few exceptions to the US "citizenship by birthright" rule, such as children of foreign diplomats working at embassies in the USA who are not born inside the embassy itself (which does not count as "US territory") but in a hospital which is in US territory.
"American-born" is the common phrase.
Firstly, it's best to think of it as as a set phrase that refers to citizenship, rather than country of birth. While you might be understood if you say "America-born," the phrase will sound odd to a native speaker, and most people will simply assume that you meant to say "American-born" and attribute it to a fluency error.
Secondly, a searchable term for the second part of compound words like
American-born is a hyphenated suffix. Hyphens have several uses, one of which is to connect modifying words, especially when needed to clarify or separate words that could be parsed incorrectly. For example, consider the sentence fragments below:
American born abroad.
Someone who is currently an American citizen but who was born outside the USA. They may or may not have been an American citizen at birth.
A person (or people, plural; the fragment is ambiguous without context) currently living or traveling outside the United States. This person was an American citizen at birth.
In practical usage, the hyphen here syntactically allows the grammatical but non-colloquial phrase
born as an American to be converted from a verb plus modifying clause into a single compound word. In either case, American is actually eliding "American citizen." Therefore, a native speaker would generally understand American-born to mean "born (as) an American citizen."
Because English is a satellite-framing language, phrases that denote motion or manner often require or imply a preposition or prepositional phrase. If it helps, you can think of American-born as meaning "born into American citizenship."
"America-born" is not a common phrase.
While probably grammatical for certain usages, the phrase "America-born" is simply not one that you are likely to hear or read. If you wanted to say that a non-US citizen was born in the USA, you would say something like
A <nationality> born in the US. The nationality would generally be expressed as an adjective like Chinese, Canadian, or Dutch (implying the noun citizen).
There are some exceptions. For example, you're more likely to hear "An Irishman/Scotsman born in the US" than something more contorted like "An Irish UK citizen born in the USA." Nevertheless, such phrases are still generally understood to reflect citizenship rather than cultural or racial identity. When talking about identity, the phrase is generally
<adjective>-American such as Irish-American, African-American, and so forth.
You'd best regard it as an idiom.
There is no logical reason why you shouldn't say "America born": it's exactly parallel to "factory made". We just don't.
Perhaps there is a distinction between country of birth and citizenship at birth? To be honest, I think it's ambiguous: I would avoid describing myself as British-born or German-born (I was born in Germany with British citizenship) because I don't think it would communicate clearly. But I would tend to assume in the absence of clarification that "British-born" refers to citizenship, not to geography: I was born British in Germany.
In comparison to another phrase mentioned; another factor to consider: "factory" is not a proper noun, but a common one. On the other side, "America" is a proper noun. The country in question is always a proper name as well, and in English, proper names cannot be modified by an article or other determinants. Nor do they modify other words.
This carries over, in that we also say "American made" or "French made", rather than "America made" or "France made" when referring to the manufacture of items. We would say "France made a mistake" to refer to the government of France doing something, but not "France-made clothing". The other difference is that in "France made a mistake" it is a noun + action structure.
By understanding them as proper names it should become clear that the supposed alternative phrases must be ungrammatical. The word "America" refers to the country itself, while "American" refers to being of or related to the country called "America". Proper names are simply not adjectives or modifiers - they refer to a specific thing. The prase "factory made" uses a common noun, which can be used in this way because it isn't a specific reference. One exception here is that you might say "Dearborne made", but this exception exists because there is no general referential term meaning "from, of, or related to Dearborne"; ie. China -> Chinese, France -> French, Dearborne -> ____.