The issue isn’t the proof of citizenship. The issue is that poll taxes are unconstitutional and there is no state that I’m aware of that makes the acquisition of identification documents free of cost.
I’m honestly quite surprised that politicians don’t resolve this idiotic situation because it’s so damn simple, but I think it’s not solved because various state governments rely on small fees for revenue. And of course because there are many political situations in which making it difficult for specific opponent voters to vote is a campaign strategy.
Make fees for drivers licenses, birth certificates, and passports illegal, and ideally institute a system that makes these forms of identification automatic/stupidly easy to acquire and the whole issue is resolved. Now you can require voters to present them and you aren’t disenfranchising anyone.