Should voter ID requirements (a la "SAVE America" act or similar) become law, constitutional prohibitions on poll taxes will lead towards every citizen getting an ID at no charge.
That's technically correct--if you show up at a place and time where the state accepts applications for an ID and submit your application with all required supporting documentation the state will not be able to charge a fee to process you application and give you your ID.
But there is no limit on what it might cost you to (1) obtain the required supporting documentation and (2) go to where you can submit your application at a time when they are accepting applications.
For older people and for poorer people these can be a significant problem. The fact is the for nearly all of its existence the US has not required much ID, meaning it is very hard to transition to such a requirement without disenfranchising many citizens. So far I don't think any proposal for stronger ID requirements has tried to address this.
Voter suppression strategy depends on statistical degradation of enfranchisement.
There is a small group for whom it is literally impossible to obtain supporting documentation — this is the group that you're focusing on, but to my mind it's not that important.
There is a much larger group for whom the difficulty of obtain supporting documentation makes them statistically less likely to obtain it — and this larger group is the one targeted by voter ID laws to suppress their votes.
Because voter suppression strategies will absolutely not be abandoned, we're on an inevitable path towards national IDs.
Many states already have required ID for voting, and indeed a state ID card can be obtained from the DMV for a nominal cost, probably free in need-based cases.
You shouldn't have to jump through hoops and prove you are needy to have free access to required government documentation. It should be free from the start because it is being made a requirement to maintain your civilian rights.