The reason that it ultimately will not stop pet abusers has already been told to you. Those that want to abuse animals will just get them some other way. You actually don't have to believe me on this. Just look at the prohibition of drugs. Has the prohibition on drugs stopped people from getting em? The answer of course is a sound and resounding NO! Prohibiting people from buying animals because they were convicted of abusing an animal will not stop that person from getting an animal. If they can't get em through those stores then they will get them in other ways.
Another problem with your registry is that animal abusers can also abuse other peoples animals, or just strays in general. After all, you don't need to take an animal home in order to abuse it. Heck, Michigan has beaches...how many seagulls you got flying around there? Know what alka seltzer does to a seagull that swallows it? Get my over all drift with this paragraph?