Voter registration is required in the US because the state and federal governments do not always track where people live. If you get a driver's license, "motor voter" laws typically give you the option to register to vote at the same time, but there are people have driver's licenses, pay taxes, or do other things to show up in government databases. Voter registration gives them a way to vote. Voter registration rolls are also used to select people for jury duty.
Localities in the US may allow non-citizen residents to vote on local candidates and issues, and some have -- mostly more "progressive" cities and counties. I think states may allow non-citizens to vote for state candidates and issues, but my understanding is that it is currently forbidden in all states. It is illegal under federal law for a non-citizen to vote in federal elections, which I assume includes presidential elections, even though technically the votes only work to select state-level delegates to the college of electors.
Maybe it would be better and simpler if we just stuck peoples' thumbs in purple ink when they vote. That would give other people a chance to police non-citizen voters (in places where it is illegal), reliably prevent multiple voting, and reduce paperwork overhead. However, that scheme is incompatible with absentee and early voting, which are considered important voting methods by both conservatives and liberals in the US.