The problem is that the system would be run by racist state governments who would construct the system in the worst possible way.
(Not a conjecture, this was actually evidence in one of the various voter ID lawsuits; the state government had constructed a table with impact assessments of various approaches, then picked the one with the worst impact)
Just because ID is free does not make it universally accessible.
If the office that hands out ID cards is only open 9-5 Mon-Fri, for example, a voter who must work during this time and can't afford to miss work may not be able to get ID.
(Not a conjecture, this was actually evidence in one of the various voter ID lawsuits; the state government had constructed a table with impact assessments of various approaches, then picked the one with the worst impact)