A friend of mine went that route for his boy, personally I think it's a good option. Another has his kids shoots 12s with reduced recoil loads, and that works for them.Your kids don't want to shoot a 12 ga pump gun. Get a 20 ga semi-auto. A Remington 1100 would be ideal, but brand-new is more than you want to spend. See if you can find a used gun. The semi-autos absorb (slow) some of the recoil in the action. You can always load it with a single shot. 20 ga. is less weight and less recoil and still gives good patterns. 28 ga and .410 are too hard to hit anything with at a distance. They might be fine for clays, but will miss or wound birds unless the shooter is a real good shot.
Personally for my kids I'm probably I'd go for semi 12s that are tuneable, best of both worlds and generally not a lot larger or heavier than a 20.
I also think every should have their own shotguns rather than all sharing one.... Just cause of fittment.