You're conflating crime and jurisdiction.

A Thai citizen can have sex with a minor and he's neither a criminal nor under our jurisdiction. An American can go to Thailand and do the same and he's a criminal, but not under our jurisdiction (he'd have to voluntarily return or be extradited). A foreign diplomat can be under our jurisdiction but not a criminal for doing the same as an American.

So an alien having committed a crime doesn't mean they are under our jurisdiction ("subject to the jurisdiction thereof").

For example, what's the difference between a criminal alien unknown to the government who is still in the country vs one who has since returned home? From the government's perspective nothing is different - until the former is discovered and so becomes under our jurisdiction - which is effective just like if the one who returned was extradited back to face prosecution here.

Also, in America only "certain unalienable Rights" come from God/nature, not all rights. Citizenship is not inalienable; in fact naturalization is delegated to Congress by the Constitution.