I find it amusing that so many people have forgotten the ultimate stamina cooldown of their own generation - "mom won't give me another quarter until next time we do laundry".
Let the kid play what he wants, but if he asks for money treat it like the arcade - limit it, meter it, and try to teach him the value of mastery. There are a fair amount of good F2P titles out there, even stamina-gated, and if he can't normally whale it up he'll gravitate toward those; at that point, when he DOES get a few bucks App Store or Play credit, he'll be torching it on the equivalent of spending all afternoon trying at high scores rather than the equivalent of feeding The Simpsons quarters until he sees the ending.
I have 2 young children and sincerely hope they don't contribute to gaming's demise (i.e. its current business models) like so many have before them, including many on these very boards. I want them to have fun and game if they want to (and my oldest who is 5 does) but like most everything at that age, they need a guiding hand. Otherwise, they may wind up as jobless adults living in my basement playing Candy Crush 5 in a credit card induced high.
Or they can end up in the basement posting to GAF "well, this Youtube guy finished The Order in five hours, but I bet he's a gaming god or he wouldn't have a Youtube channel so it'll probably take normal people like me 15! And 15 hours of Epic Content® is worth 80 bucks, so it'll be a great game!"
F2P's so hot right now because it brings all the bad AND the good of the arcade model to a userbase that finds something vital missing in a Skinner's-rats-style chase after the next cheevo or cutscene. For the medium, or at least the less-than-a-medium more-than-a-genre, to reach its full potential, we need to reject BOTH the bad-cop Skinner boxes, modern-day quarter munchers like iOS Dungeon Keeper that tell you you can only get an endorphin rush by dropping in another credit, AND the good-cop Skinner boxes, content tours so common in AAA that promise you the endorphins will never stop flowing as long as you always have fresh content to database as Azuma described.