That's true, but some people really have very little to offer to the market. Too little to support themselves economically. That's the sad truth. And unrelated to that, minimum wages are a tax on dumb work. I like that.
I guess we agree that one should make sure everyone has enough income to live. Something like basic income (or so) might do a better job than a minimum wage.
I often hear basic income heralded as a sweeping reform. Perhaps one should try to position basic income as relatively small step to contain bureaucracy: All means of state welfare (and we have a lot of them in e.g. Germany) will be combined into one; and as the state already knows how much money you make for tax purposes, it just re-uses that knowledge to determine your basic income allowance.
But isn't the problem exactly that only a sweeping reform - i.e. basic income replacing all the other benefit schemes - will actually lead to the kind of reduction in red tape that makes the system pay for itself?
Obviously, this is a rather intricate matter and very dependent on the particular welfare system you're talking about. I don't know anything about the german one.
But I think, politically, the big issue is the idea of handing over money to people without requiring them to look for work. That's a tough nut to crack, no matter how convincing your anti bureaucracy stance is. I doubt that it's going to happen.
Abolishing the minimum wage does not mean, that suddenly all former minimum-wagers have to work for free.