> the mentally challenged, the old, the handicapped
We have disability safety nets (that could be better). And FLSA and state regs all allow paying these groups under minimum wage (and this is a good thing).
My wife's brother is severely mentally handicapped. He lives in a nice, well-run group home (I'm aware this is not universal). He also has the option of going to "dayhab" where there's educational content and the option of working making crafts for a couple of dollars per hour of spending money.
I am not sure his labor output is worth the $2-3/hour he gets paid: a lot of this is probably charity, but it makes a difference to him to be able to earn something and work towards goals, and not just spend his entire day watching TV instead.
You don't fix these things with a minimum wage and you possibly make them worse. They've got different solutions.
> The young, ... the convicted
These are harder problems.
* How do we provide a good on-ramp to productive work and independent life for at-risk youth?
* How do we allow those who have broken the norms of society in the past to rejoin in an orderly way, with hope and better prospects from compliance than reoffending?
Note that I think we should have a minimum wage. I just think it's also a band-aid at best, and often causes a fair degree of problems as well. It's better than nothing, but one would be better off addressing the problems it's targeted at in any other available way.