Frankly I think that level of restriction if bullcrap when applied to what is a plainly written right but this level of restriction is applied piecemeal to various rights by various states both at present and in the recent past. Legislatures does not see things as plainly as you or I do. They deal in these kinds of mental gymnastics as a routine part of their business. Just because something's obviously incompatible with the constitution doesn't mean it can't be the law of the land (especially at a state level) for a generation or three.
As an aside, an ID is typically required for all sorts of things that poor people do a hell of a lot more than rich people (change jobs, use a check cashing service, sell scrap metal, sign a lease on a new apartment, register a car, apply for .gov assistance, interact with the government in general, etc.) so they do tend to treat acquiring/renewing IDs with more urgency than the HN crowd because not doing so will cause them problems in shorter order.
The threshold where "public safety" is considered with respect to limiting individual rights is certainly up for debate in various cases - including gun ownership. That said, I think there are pretty clear differences between the public safety risk surface of "voting" and "owning a firearm", so I think it's not reasonable to assume that they should require the same standard of restriction (i.e. in the form of tracking/ID requirement).
1) knowing enough about the process to find all the documents required to get an ID (usually includes birth certificate and SSN card if you don’t already have an ID, which not everyone is able to keep track of)
2) taking a day (or multiple days) off work to go to the DMV, depending on wait times and if you’ve brought enough documentation
3) paying $40+
4) waiting days or weeks for the ID to come
All for what? If you don’t have a car, and you don’t really interact with the government or banks in any meaningful way (you would be SHOCKED how many people are in this position) you basically don’t need an ID at all to function. So if the ONLY reason to get an ID is because it’s required to vote, this ends up being a MASSIVE hurdle. All that effort just to vote? And you need to do it many weeks before the election? Of course that will suppress turnout from populations who don’t already have IDs!
This isn’t even considering populations who have pretty deep-seated (and reasonable!) wariness of giving too much information to government institutions, when historically that information (especially demographic) has been used to oppress/discriminate in things like housing policy, policing policy, etc.
If America issued a federal government ID at birth for free that auto-renewed, maybe these laws wouldn’t be as insidious. But that almost certainly will never happen.