This is a really great question. I think it helps to back up and see the big picture once in a while. What are we trying to solve, is it truly a problem, and can it solve the problem (or at least reduce occurrence of the problem)?
I don't think these laws would solve the problem and I also don't think it would suppress voter turnout. It doesn't sound like there's a ton of fraud (there's potential, but I don't think it has materialized). It also doesn't sound like there are very many people at all who don't have IDs. Otherwise they couldn't drive or buy alcohol or tobacco or do taxes or rent apartments or get jerbs or do any of the other things that kinda require ID.
Yet here we all are arguing our asses off about it for 11 pages

Sometimes I think the powers that be
want us arguing about stuff that doesn't matter.