You don't want cashiers, wait staff, grocery baggers or call center staff? All of these jobs are generally minimum wage.
Low skill call centers are total waste of human time, that's why most of it is outsourced. UX just can get improved instead.
What if they are willing to work for less because it serves their purpose (like a foot in the door in the industry, or something else that you and I can't even think of). For example, we allow unpaid internships because we see the value of it. What about an internship that pays a little? Isn't that better for all?
The problem is you're taking choice away from people who may want it or need it. Why do you think you know better what's good for them than they do?
Basically min wage should be defined as basic needs for individual. I don't think that many people will chose to go for lower than that for future personal gain.
Anyway that's what democracy is for :)
EDIT: example: you don't provide living wage so some/more ppl will chose to steal. This might destroy their family, so social worker is needed, you also need more police, judges, prisons etc. All of them are employed, happily contributing to GDP. Success! Is it better world? I don't think so. It's much cheaper if with offering living wage. (as global optimisation, company owners and individuals do local optimisations)
Just because a few people are willing to work for less - or for free - to get in the door doesn't mean it is just to take advantage of them and pay them less than a living wage. Not everyone will be working x job for opportunity. Some folks need to eat. And yes, it is taking advantage of people - artists and musicians get offers to get paid in Exposure all the time. Which is really, "please work for free!".
The only real exception to this is, perhaps, when the work is actually part of schooling, and that should be limited. Teachers, doctors, and pharmacists come to mind, though we do treat the doctor interns quite horribly. (some pharmacist interns get a bit of pay).
If you want labor, pay a decent wage.
Even after a minimum wage increase, the school will still have to vacuum the floor. I'm guessing it'll just have the other staff do this now, who aren't children.
I can't verify they feel good about getting a paycheck. I have generally been happy to have a bit of money, but I've never cared that it came from a paycheck and realistically, despised that I had to work while going to school (mine was later, in college). I'd have went without if it were a reasonable option.
I realize that folks are different. However, In a lot of places and situations, child labor is illegal - and 14 is considered child labor in so many situations. When they do work, it comes with massive restrictions. This is soley because we, as a society, generally feel that the focus should be on school instead of work. If we cannot pay young folks on par with adults, we shouldn't hire them.
As a sidenote: I'm guessing that the real reason the school hired them is because they are cheap labor - soft exploitation - and so long as their rate is less than paying a member of the custodial staff (who would have to pick up the slack), they won't be out of a job.
I assure you that these teenagers are not being mistreated, and there are adults doing this same job with them. Our society has come a long way. No teen spends 100% of their time on school work. Like anyone, they have a price they are willing to trade their time and labor for. Why should they, or anyone not be allowed to make that trade?
And yes, it is an emotional argument. We are emotional beings. Doing a useful job and being rewarded for it satisfies those emotions immensely.
In addition to that, it teaches things in ways that a classroom simply cannot. In a very real way it is paying them to get some education, as you suggested.