Being a player and being a manager are rather different things. And the whole concept of "schooling" suggests a manager coming into a club to continue that clubs ethos. In most cases here - not actually what happened, is it?
Guardiola the obvious example, though he took the essence of the club and evolved it.
Klopp was an up and comer who went into a massive club in Dortmund and made an immediate impact, having had nothing to do with Dortmund prior.
Similarly Jose with Porto. And even then, while he had had that success at Porto - Im not sure anyone anticipated he would do with Chelsea what he did.
Conte left Juve and managed minor teams, was seen as a young talent and did incredible things at Juve. Being a player, I would imagine, only serving to please the fans. I doubt it does anything to counter the point I raised - young, up and coming managers, can go into big clubs and make that step up. It is far from the requirement people seem to believe that experience is a necessity.
And this was but one point of many, but seemingly the only one you wanted to zero in on.