Well, there's no ID law that will stop me from getting on a plane and voting in Salt Lake. If someone were hell bent on it, there's really nothing keeping someone from making a fake ID - it won't take much to fool the retirees that are typically working the polling locations. Actions that have been taken before to straighten up the registrations tend to scrub out plenty of legitimate registrations.
Interestingly, I don't see that there is any mechanism for me to tell Salt Lake County I don't live there anymore and to take my name off. There is no de-registration process.
I don't think there is anything preventing states from figuring this out on their own. They could do it through the website I used with a REST api and a tiny python script. If you have the information in one database, you can compare it to someone else's. No need to pass any amendments, or to federalize voter registration, which would be impossible anyway since different states have different standards.
I dislike the characterization that people can't effectively articulate why voter ID and extra hoops are an issue. You can reject the arguments, but they have been coherently made.