The problem I see with raising the minimum wage is that it also causes price inflation. It is kind of self defeating. Raise the minimum wage and people who make more than minimum wage actually end up with less purchasing power.
But the effect is not the same for all.
Yes, all things get more expensive, but not by the % of the wage increase. For once not all work is done by humans. So if 50% of the work is done by humans, a 2% increase in wages means only 1% increase in costs (yes, its more complicated, but just to illustrate).
Also the increase gets paid by everyone. Even millionaires. So a rising minimum wage is a better redistribution of wealth.
I understand what you are saying, but at present, you are making some pretty big assumptions. I can not think of any minimum wage occupation that is even remotely at 50% automation. I personally am not a fan of wealth redistribution. However, I am a fan of wealth creation and I think that is where people should focus.
Just compare getting potatoes out of the soil now and 100 years - heck 50 years - ago.