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.
The plan is not coherent. Some items to consider:
Who is verifying the documents? If the names have to match, what about people who change their names? What documents should they present to prove that that's their real name? How will the election worker actually verify that the documents are correct?
Women who have changed their names after getting married have a higher burden of proof than people that have not changed their name. The folks who wrote this act are aware of this.
The SAVE Act, in particular, puts in place criminal and civil penalties for election workers that fail to properly identify someone. But a random election worker is not equipped to make that judgment perfectly and is going to end up making mistakes, since document verification for things like birth certificates is completely manual.
Young voters are more likely to have the documents like a birth certificate back at their parents' house and they're not likely to yet have a passport. Likewise, poorer voters do not have the resources to easily obtain these documents. Passport costs over $100 in the United States and for someone that doesn't travel often is not a very good investment. Just because you can't afford an expensive government document does not mean you're not entitled to your vote.
If Silicon Valley has taught me anything, it's every time you add friction to a process, fewer people have the resources and/or are willing to go through it as they drop off the funnel. In this case, it's targeted at the young, the poor, and the women. We used to be a nation that would say: get out and vote! Participate in our democracy! Now we're looking for excuses to not let citizens vote if they can't prove on the spot that they're a citizen. Papers, please.
And all that for what? Multiple institutions, including the Cato Institute and the U.S. Department of Justice, have found no evidence of meaningful voter fraud. This is because identification is required during voter registration, which is effective in preventing illegal immigrants from voting. This is why we tie voter registration to things like getting a driver's license at the DMV. You already have all your documents prepared that you gathered for that visit, and so that's the right time to be checking all of them.
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.