I think you’re making the mistake of assuming that this thing that we can’t really verify (because we can’t make sure <person voting> = <person registered> at the polls) isn’t happening, precisely because we can’t accurately verify it. It’s not a theoretical concern that voter rolls can be stale (because of not removing dead people or people who have moved in a timely manner) or otherwise inaccurate. And attempts to actually purge voter rolls always meet stiff resistance as some nefarious ploy to disenfranchise voters. There is at any time a non-zero chance that you could vote using the name of someone who’s either dead or not around any more. So why so much resistance to safeguarding against that? Nevermind the added benefit that a national ID card could be used as a real replacement for Social Security numbers. But again, so much resistance to something that every other country thinks is a good idea. Which is even more assuming since we point to “well everyone else does it that way” for so many issues. But voter ID? Oh, well that’s complicated, couldn’t possibly work here.

They meet stiff resistance because they're always done at election time and only selectively.

Voter ID laws are a non-starter because historically they've been used, along with literacy tests and civics tests, to disenfranchise people who can't get an ID. For example, in Idaho you must have "proof of your identity and age" like a birth certificate or citizenship certificate, plus proof of residency like a utility bill or rental agreement or employment record.

These things are easy for most people to provide, but people who are in unstable living situations may find these things impossible to provide. Requiring those people to provide ID at the polls would effectively disenfranchise them.