I personally like the 410 approach. I asked the question about 410s a few years ago and got both sides of the story, many people discouraged it. I went ahead and bought a Mossberg 510 mini. Yes, it's tough to shoot flying objects with a 410, however, a young kid can learn lots with these guns. Such as, proper gun handling, gun safety, gun ownership pride(you should have seen my son's smile every time he laid eyes on his gun), cleaning and maintenance(my son took a big interest in cleaning when it was his own gun), and the list goes on.
We went small game hunting quite a few times, mainly squirrel. My son still talks about the first time we got a squirrel and made squirrel stew.
I started him out with stationary targets on the ground(clays, pop bottles, cans, etc) Once he had that figured out, we moved to rolling targets. I bought a case of rabbit clays, then I would stand right behind him so they would roll straight away from him, so he was only dealing with one axis. Then we moved on to diagonally moving rabbits. For flying targets, we started with Frisbees(seems like we always have lots laying around from parades). Frisbees are big and slow, so they have some success with these.
So no, a kid won't be knocking pheasants and geese out of the sky with their 410, but there are so many other things they can do with them. We have countless hours of quality bonding time at the range. I also bought a reloader and the kids love reloading too. So more bonding time there.
If I had to do it all over again, I'd buy that little 410 again in a heartbeat. He has outgrown it, but our daughter will be growing into it soon, so it will get more use.