Earlier quoted context omitted.
Any ballots that are cast under same-day registration are cast as provisional and will go through the full verification process if the election is close enough where those ballots are necessary. Source: actually ran a fucking election precinct. Non-citizens aren’t casting ballots illegally.
They literally just charged someone in Philadelphia for illegally voting in every federal election since 2008. Non-citizen, ordered deported back in 2000 but still in the country. There's not been a reliable audit to show the extent to which this happens (probably not enough to affect even local elections), but to say that it isn't happening is just a lie.
What they say is that it does not happen enough to plausibly come close to affecting the outcome. And this is widely supported, including by right-wing organizations (as a sibling comment observes).
As with most issues, there is a trade-off here. As you tighten controls to prevent improper voting, you prevent some people from realistically being able to vote (it's just too hard, time consuming or expensive for them to meet the documentation requirements), and discourage others. This is particularly bad for the 1-2 elections after the rule change, which most people won't know about until they show up to vote. IMO, this is really the point of the changes.
And you have to weigh that negative against the supposed benefit. But that benefit is really hard to find. It's very clear that intentional voter fraud (fraud in registration, or in-person impersonation) is extremely rare, and does not come anywhere close to affecting outcomes. It's already a crime, and we seem to be pretty good at catching it.
The other argument for a benefit is that it improves voter confidence in elections. I reject this, since the only reason the public at large has any real concerns is because of intentional misinformation by the right. You can't lie to people to convince them there is a problem, and then use that to justify your heavy handed solution.