Your vote is tied directly to where you live here, all political offices cover some territory (a state, city, district, village, town, etc). So your vote is directly tied to where you live.

But it's not necessarily the case that the state always knows where someone lives. Yes, there are property deeds and utility bills and what-not, but there is no official place where everyone's addresses are held.

Therefore, "Registering to Vote" is really more like registering your address, so they know which ballot you need to vote on. Even ex-pats what no longer live in the country (but are still citizens) get to vote at the last US address they lived at.

There isn't always an "unregistration" process either. When you move, you just re-register. Since most of these are administered at the State level, if you move between States you may end up registered in two places. In most cases, that is perfectly legal! You only break the law if you actually vote in the same election in more than one place.

One of the major parties wants less citizens to vote, because they do better if less people vote, especially "those people". The other of the major parties wants all citizens to be able to vote. So setting up an automatic voter registration in all states, since this IS a given state's right, is a bit of a conflict, oddly enough. I absolutely agree it would in fact save a lot of trouble to just register every US citizen at age 18.

elections in the US are run by the states, not the federal government. a handful of states (specifically California, Colorado, Hawaii, Nevada, Oregon, Vermont, and Washington) automatically register eligible voters and send a mail ballot to every registered voter

We register to vote in other countries too, it's just an automatic process that happens alongside other things like filing your taxes.