Because the name of every person who votes is recorded (at least in my state, and I'm pretty sure that's every state to detect double-voting). That's how, incidentally, the parties know to hound you if you stop showing up.
It should be very straightforward to compare the purged names with death certificates with voting history; a state government should have access to all that data, though it may cost money to collate it.
Did Texas do that? What did they find when they did? I feel like if they'd found something we'd have heard about it by now.
Dead people still being on the voting roster has never been a problem because, in practice, the amount of fraudulent voting under another person's name is always low. Do we have reason to believe that has changed?