> there's absolutely no evidence that they've ever voted more than a random one-off that's almost always caught and prosecuted.
To be more precise, the evidence is that non-citizen attempts to vote are somewhere on the order of one vote in ~hundred million. Screwing up voter registration forms is somewhat more common, but people who actually cross reference databases have found that the people who incorrectly mark "I want to be registered to vote" on their drivers license forms don't actually vote, which suggests it's mostly a mistake.
There are issues with the amount of people on voter registration rolls who shouldn't be, but I should also point out that anytime someone claims ridiculously high numbers of potential misvotes, it's because they do coarse-grained database mismatches, and so they'll catch people whose names match. When you do the actual legwork, the voter registration issues go down dramatically, and the principal cause of that is that people aren't correctly unregistered when they move (and people don't attempt to vote in their previous place of residence, so there's very few actual vote attempts!)
In fact, the largest category of illegal vote attempts I believe is felons who try to vote. Because the voter registration system does a poor job of actually working out who is a felon whose right to vote has been revoked.