I am italian, I see mafia people obtaining positions of authority (governor of a region, mayor of a big city, leader of a nation-wide company) every day. So sorry, but you made the Giovanni example with the wrong guy.
Concerning the kids, you are not addressing my point, but rather a small portion of it (not that I am surprised, you do that all the time when we debate Se7en stuff, too). I never said that I have a problem with their age. I did say that I have a problem with the fact that they are locked up in an orphanage, and not only is it openly established by the game that they are forbidden from having any contact with the outside, it is also established that, when a gym leader is no longer eligible for their role, the reserve gym leader gets it instead. Maybe I am being overly nitpicky, but "being locked up in an orphanage with no contacts with the outside" sounds like a strong case for being "no longer eligible" to me. So even if Charlotte got her gym by being her mother's substitute, I'd still expect HER OWN substitute to have been named (and to have taken over) by now.
Noel finally manages to have a gym battle, against you, after you break him out: had he remained in the orphanage like he was supposed to do, with his Pokemon taken from him (and, again, NO FRIGGIN' CONTACTS WITH THE OUTSIDE), how was he supposed to have gym battles at all? How was he (and the other kids for that matters) supposed to honor not only the "double responsibility" part of his job, but even the most basic duty that comes with it?
And "Reborn City is screwed up" is not a satisfying answer: that only explains why the kids would be locked up in the orphanage and abused by Sigmund, but it still fails to explain why a system established by this very game, that worked perfectly in other occasions, would not work here. If you want the fact that the system failed, and the kids not only landed the job but actually kept it despite being locked up in an orphanage, to be a plot point, you need a MUCH better explanation for it. Also because, in this screwed up society, Sigmund is actually regarded as a pillar of the community and a brilliant scientist, allowing him to get away with his horrific experiments... Which goes to show just how screwed up Reborn City is, yes, but it also causes another issue: Sigmund is seen stating, in multiple occasions, that the kids have serious disorders and to be unreliable at best and lost causes at worst. To the point that he sees fit to take away their Pokemon from them. Then wouldn't it follow logically that he would also recommend them to be lifted from their positions? And wouldn't his word be trusted on the matter, as part of the whole "Reborn is screwed up" thing?
So yeah, I stand by my earlier points. You say "we still don't know enough", but see it this way: maybe Ame, upon reading discussions like this one, will feel motivated to address the issue in a satisfying manner in future episodes. Isn't that the point of "constructive criticism"?