The reason a minimum wage is necessary is that supply and demand of labor has been skewed. Invaders (legal and illegal) depress wages at all income levels. There’s no protection from unfair completion against goods from slave labor countries. The irony of that is a lot of high-end merchandise is made in these conditions, and that merchandise isn’t any cheaper than it could be if it were made in the US. Corporations, celebrity endorsers, transportation companies, contractors who secure the sources....they’re the ones profiting.
I recognize the problem, but don't agree with your solution. An important reason invaders can work for below minimum wage is that they receive other benefits from the government. In other words, hiring an illegal who works under the table for sub-minimum wages is cost shifts costs from the employer to the taxpayer.
I don't see how a minimum wage helps that, because the illegal still gets more total benefit than the non-illegal who is just getting straight wages.
There is nothing wrong with minimum wage as a starting wage, or for teenagers.