I'm not entirely sure mandatory voting is wise in a country the size of the United States, but I'm with you on the voter ID and the rationale. I was shocked to learn that with everything else that requires photo ID, it's not required to vote in American elections. (At least, in some locations; laws vary from place to place). That just looks bad.

My personal opinion is that if people think they can benefit from cheating, a certain number will do so. It's just human nature. We're seeing more and more of this with LLM cheating on the rise in universities. So there's always going to be a certain number of people who want to vote fraudulently — after all, if the right person gets into office, it'll probably benefit you. Your taxes might be lowered, or your government handouts might be increased, or whatever other reason you have for preferring one politician over another. Preference alone does not mean that people will vote fraudulently: after all, most legitimate voters also prefer one politician over another. But the easier you make it to cheat, the more people will cheat successfully: among those who wanted to cheat but didn't, usually the only reason they didn't is because they couldn't see how to get away with it.

So photo ID to vote just seemed like common sense to me, along with other measures like ballot boxes kept in a publicly-visible place until they're opened, to help prove that nobody has tampered with them. (Read up on the 1946 elections in Athens, TN sometime — there, the fraud was being done by the sheriff and his cronies who would take the ballot boxes away, "count" them in private with no outside observers present, and announce that surprise surprise, the sheriff's crony had won the election again).