On the one hand, it seems only common sense that voters prove their identity by some government-certified document before casting their vote. Provided of course that such documents are not unduly hard or expensive to obtain (it would, needless to say, be a lot simpler if you just made those documents universal and compulsory as they are here, but I guess that's not to Americans' taste).
in general, you typically have to show a recent utility bill in your name, at the address you are registering to vote from. barring that, it is up to the state you reside in to prove you both are a resident for enough time to be eligible to vote, but also what that proof entails. requiring a mandatory ID would make a lot of things like this go by the wayside, but you are correct that enough Americans believe this to be a government intrusion into their lives that a national ID has never gotten any traction
with 30-ish cases of in-person fraud over billions of votes cast, i'd say the current system is doing a decent job of preventing in-person vote fraud. the biggest problem is that in all of the republican-controlled states enacting "reform", the obvious goal is suppression of left-leaning voters in order to consolidate power for republicans. none of the republicans here will ever admit to it, though, because that would cause them to think about how their party is failing everyone, and thinking is difficult for most of them
"That's the trouble with political jokes in this country... they get elected!" -- Dave Lippman
