Without a real-time, probably national "who has voted & where did they vote" database - how would a "just show up and vote" system block a citizen from voting once in each of multiple jurisdictions?

You can solve that simply by putting all such votes inside signed envelopes, and waiting to count the contents until all the envelopes can be checked for duplicate voter details. In Australia this can be forced on you if you are caught double voting, or you can opt into it if you don't want to appear on the (public) electoral roll (and hence can't be ticked off), or you're voting from outside your electorate (so they don't have a copy of your electorate's roll).

Hard enough to get a person to vote once. Probably not a big problem.

The big problem is, the folks who count the votes and cheat. They can invent an arbitrary number of votes to swing their guy.

Let's worry about the problems that matter.

SSN? To me this is one of those friction things, why is it hard? Like taxes, I would take an option rather than tallying up just pay a flat $5K fee or something under your expected tax bracket.