You are arguing that dishwashers get paid too little. Your argument is that dishwashers produce a lot of value and you're right about that. The restaurant is willing to pay probably as much as $50/hour for a dishwasher if not more. The problem is that people are willing to take the job at $15/hour so there's no incentive for the restaurant to pay more.
People mostly get paid just the right ammount according to the market, but from any other percepective the numbers are pretty aribtrary and by no means justifiable. At most one could say that some people don't work hard on their carrees and e.g. try to make a living from beeing an artist (or in oter words: don't optimize for money). But this is not true for the dishwasher either.
One guy which I knew, a movie FX guy (Maya, After Effects, ...) moved to London and washed dishes for a few months before getting a gig in his field.
And I've met quite a few more people in London doing very low paid jobs.
They are paid so low because you can easily find someone willing to do it for even less.
I'm not saying that this is fair, it's just how it is.
Basic income, which is inevitable, will fix most of this, as in nobody will need to work just to make a living.