I mean, minimum wage doesn’t “make sense”, because it’s a price control. It’s effectively a bundled subsidy to the worker and a tax on the employer. The subsidy part is fine, but:
A. It’s a fallacy to link specific taxes with specific subsidies. It’s called “hypothecation”.
B. Even if you’re going to hypothecate, you can at least tax wealth or a negative externality instead of taxing the random person who happens to be the recipient’s employer.
The obvious solution here is a Universal Basic Income (UBI) instead of minimum wage. (Note, Germany only got a minimum wage in January of 2015, so if you think it sounds crazy not to have a minimum wage, it’s not.)