Wouldn't the IDs allow voting centers to better keep track of how many people visited the location and prevent ballot stuffing happening later in the process?
Sure! Each precinct records the number of people who physically came in to vote, how many physical ballots were cast in person, and those should match 100% More ballots than people means something fucky's going on, while fewer ballots than people means either a ballot was deliberately spoiled (someone walked out without turning it in, or otherwise changed their mind and had it "voided") or went missing (also considered fucky).
The trick is that for 30+ years Republicans have been literally barred (physically, by court decree) from observing the in-person process or otherwise inspecting those low-level numbers. Not even kidding (it's been discussed elsewhere in this thread; fortunately this consent decree
expired in 2018 -- note how the article says this is bad somehow), which is probably going to make today's elections "more interesting" than usual since these games are going to be harder to play).
So any democrat-leaning precinct is free to just turn in more ballots than they had registered voters. Combine that with the rampant mail-in ballot fraud, and you've got easily-cheated precincts. Yes, the numbers are blatantly fraudulent, but, unbelievably, not only could Republicans not monitor any of it,
they were forbidden from suing over it when they suspected (or could prove) the numbers were wrong and the precincts were lying.
So yeah, you could easily prove fucky shenanigans by looking at the numbers. If you were a Republican, you just couldn't do anything about it when you found it, and there were legal roadblocks in your path even trying to look.
This is why they're fighting voter ID so hard. That decree is gone, so now we fucking
can check the books and sue when something's amiss. Without voter ID, they're free to just register everybody in each precinct if they want (and even possibly non-existent people) and then vote on behalf of anybody who doesn't show up and mysteriously hit 80-90% voter turnout levels in key districts). If there's a verifiable paper trail between each registered voter and the total number of times each registered voter voted (i.e. should be "0" or "1"), you can't cheat like that. Without it, you can screw around with things quite a bit.
Mail-in ballots need to be banned too except for overseas military personnel, and even then it shouldn't be "mail" but fucking armed military transport that brings those back in for counting. It blows my mind that nobody was shot (in Fortnite) over the multiple instances of "woah neat I just found exactly 2,000 fresh-printed ballots in the trunk of my car meant for this district!" bullshit. That was so fucking blatant.