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If they don’t jump through whatever hoops they are supposed to to prove they are citizens, then yes, they will be removed from the voter registry. I don’t know if you’ve ever had to prove you are a legal citizen before, but it isn’t super easy. It takes time and money to get the proper documents.
There are plenty of states where you can vote without having to show identification at the time of voting. You do have to register prior to voting, though, and that process requires identification and proof of residency. And no, none of this has ever resulted in any significant voting fraud. Between 2000 and 2014 there were only 31 documented cases of voter impersonation in the entire country.
https://ballotpedia.org/Voter_identification_laws_by_state