Your argument here is based on there being an infinite amount of labor available. This isn't the case. It never has been. I've talked previously about the difference between England and India just prior to the Industrial Revolution. Employment is a bilateral relationship. It's not just a race to the bottom by employees, but a race to the top by employers, because there are multiple employers for employees to work for.

The limited amount of employees makes it so employers have to pay competitively. When there's a surplus of labor this obviously isn't the case but the minimum wage doesn't help that problem, only aggravate it by pushing more people into unemployment be making it illegal to work for a certain amount. Therefore if your labor is worth less than what the government says is the minimum, you don't eat.