In the gastro industry it’s extremely common (at least in my country) to be employed illegally (as in cash in hand directly).

Most restaurants you go to to eat would have to close if they needed to employ their workers legally. Same thing applies to bars and similar venues.

Source: worked in that industry 10+ years

Compensation for work comes in the forms of money, working environment, flexibility, etc.

A minimum wage is a ban on certain baskets of compensation for work.

It doesn't prevent 'exploitation', whatever that means. Companies fiercely competing for labour is what you want here.