I'm familiar with the US federal system, but as far as I know, the individual state IDs still have the same problem. In particular, they're apparently difficult to obtain for poor or disadvantaged people, so there are enough people without an ID to let the insecure ad-hoc systems exist in parallel. So something like knowing the SSN, or displaying an utility bill (trivially faked) exists as a parallel ID form.
European government-issued IDs don't work well just because they are accepted, they work well because no other ID is accepted, and that's only possible when 99% or more of the population has such an ID (and the rest can be handled in a somewhat more convoluted but uncommon procedure).
Of course I have no good insight into how feasible it is for a US state / federal government to ensure that everyone (for sufficiently large values of everyone) in the state / country has an ID, without disadvantaging anyone.
European government-issued IDs don't work well just because they are accepted, they work well because no other ID is accepted, and that's only possible when 99% or more of the population has such an ID (and the rest can be handled in a somewhat more convoluted but uncommon procedure).
Of course I have no good insight into how feasible it is for a US state / federal government to ensure that everyone (for sufficiently large values of everyone) in the state / country has an ID, without disadvantaging anyone.