Look, I understand why this is happening. It's part of a concerted effort to make voting harder, and challenge the eligibility of certain classes of voters who tend to vote a certain way. But we do have eligibility requirements for voting, and the logical time to check them is when registering.
The real question is whether someone who has no documentation whatsoever should be disenfranchised. Like that homeless guy in the 59th Street subway station. He says he was born in Brooklyn in 1966. If that is true, and he is a citizen, he is eligible to vote whether or not he has a pristine copy of his long-form birth certificate. We need to have a system to accommodate him.
And understand what the end-game of Republicans are. They want to couple this with an aggressive purging of voter rolls. So they maliciously un-register people who they think will vote the wrong way, then impose these paperwork requirements on these people, all with the goal of discouraging them from voting.
So, while it may seem reasonable to demand proof while registering, that reasonable request is part of a larger goal of disenfranchising large groups of people.