

Verification of citizenship is already done, the burden just isn’t on the citizen. You have to provide a lot of information to register and if it doesn’t check out you won’t get registered or you’ll be arrested if you falsified something. Successfully registering when you’re not a citizen requires either big lapses on the govt side or extensive personal info from a US citizen. This is why immigrants voting, legal or illegal, is incredibly rare.
I don’t live in bumpfuck nowhere, I live just outside a major metropolitan city. My wife had to wait 3 months to get her DL renewed and she had to drive over an hour to get to the office, IN A DIFFERENT FUCKING CITY!