Agree that it's a pity not being able to apply for MPC if below 18.

Alcohol is one potential reason, but I think as well that this could be easily mitigated, e.g. by denying lounge access if unattended as already posted.

I think as well that it's rather related to cost issues. How many kids below 18 are actually making decisions and paying for their flights on their own? In many countries, contracts made by minors can be even reversed if not approved by parents. It's usually still the parents who make the final purchase decision.

So when strictly viewing MPC as a loyalty program, then minors may not be that critical/essential to the key objectives of the program when it comes to saving costs. Sure, in case of OP a family would decide to rather give more business away to other airlines. But how many cases are there, if parents are not already purchasing J or F tickets for their kids anyways? Saving costs makes most sense where it hurts least, and that is often a combined factor of severity and frequency - either one is low or even both when denying minors to join MPC. Whereas business travelers and some other groups may be more attractive to lure with benefits and for MPC/CX to pay for those.