The CHL club would try to keep the 3 overagers they feel would be most valuable to them (or possibly not, if they're rebuilding). For the rest, they would have to find takers among the other clubs, which can be difficult, since of course they too have limited spots. As mentioned above, those that can't find takers will have to transition to junior A, or CIS, or out of hockey altogether. Having 10 in the same year is a bit exaggerated, but teams making a run will sometimes load up on older players, which can put them in that kind of situation heading into the offseason. Still, teams rarely enter the next season with more than 4 overagers.
As for the NHL clubs, beyond whatever influence they wield behind the scenes, they have no real say in this. If they're not transitioning their prospect to the pros, they'll probably push for him to play in a prominent role on a competitive team. And if the prospect can't find a spot in the league as an overager, well he likely doesn't have much of a future in the pros regardless.