If the intention was to actually fight current or future voter fraud then they should really have an easy way of submitting that documentary proof. I don’t know what form that would take in the US and how expensive it would get, considering you’d want some stipulation that it shouldn’t have much cost to acquiring such documents, shouldn’t be too difficult and whatnot. Assuming you’d want to do that right. Not that I think that’s their genuine intention.
Where I live in Finland we don’t have voter registering. We do check IDs when you vote, that part just seems sensible, but there isn’t an actual ID requirement. You just need to be identified without a doubt, but the form isn’t set. In specific circumstances it could even be that the officials there know you and guarantee who you are. But if you don’t have a passport (rare not to have it here), you don’t have driver’s license or ID card, you can get a temporary ID for free from the police station just for voting. But then you need to be also somehow identified there, so sorta the same problem again, but at least you have more time there than in the voting place.