The problem with this is that in the U.S., people do not have a “registered address” or anything close to it. Many people have multiple addresses, either temporarily or permanently.
Two common examples would be university students and military service members. Both groups have their “hometowns” and their current addresses, and can legitimately consider either to be their primary address for voting purposes. For instance, if a soldier orignally from New York is stationed in Georgia, it wouldn’t be fair to require him to vote in Georgia as he may consider himself a New Yorker who is just temporarily out of state in service to his country. On the other hand, the soldier may have enlisted to get the hell out of New York, and may consider Georgia to be his new home (or perhaps his new home of the moment), and be much more interested in and informed about Georgia issues, so it wouldn’t be fair to prohibit him from voting in Georgia, particularly as he may be a career service member who hasn’t returned to New York in 20 years.
There are plenty of more examples of people who, for whatever reason, don’t have a single permanent, registered or fixed address. Moreover, even people with a single residence may move, and there is no central (or really other) registry of where they live. I have known plenty of people who have moved to New York, but have not bothered to change their prior state drivers licenses until they expired, and even then have sometimes renewed them when they were back in their hometowns for the holidays and the like. Similarly, if there were some sort of registry like that, there would be plenty of people who would want to have registered addresses other than where they actually spent most of their time for reasons good, bad and neutral (how many people have we known who “officially” lived in one place but who actually spent every night with their girl/boyfriend).
So, people in the U.S. get to register to vote voluntarily where they consider their residence to be. That means that there will always be questions of whether people are actually legitimate voters. There will likewise always be a balance between making it easy for potential voters to register so that every vote may be cast and making it difficult so that those with marginal claims to eligiblity in a particular district may be weeded out.