Cant complain about my axis, factory mounted (chinese) weaver 3-9, for about $350 at rural king. It shoots to the quality of ammo you put in it.
Very light and handles well, I really love it for that.
As a typical to above average size man, recoil, even prone, is nearly nonexistent with the gigantic cushy rubber pad. Did a 25cent trigger job and opened up the stock with some sandpaper on a dowel to ensure it's floated.
It comes with a stock spacer installed, but I fear removing it still won't be enough for your kid to comfortably handle, as well as the (IIRC) 22" barrel hanging out there. Maybe if it was cut down an extra stock-spacer length, and barrel shortened to 16-18".
I'd say as it is, the axis might be better for your son when he's 14-16 if you're both small.
Is an AR pistol with the gas shut off an option? Or get a single-shot sled or blocked magazine...
I don't think the american rifles will be much better in the handling department, but I would expect a much better fit/finish and accuracy out of them. My american rimfire is miles ahead of the axis.
My wife is 5'2" 125lbs. She isn't comfortable with anything much bigger than a M4. Usually fits youth model guns perfectly. She finds merit in the sizing of the axis, but would definitely want the spacer removed if dedicated to her. AR pistols fit her perfectly. Gets by with her model 500 20ga youth OK, I think the set-back forend that mossberg puts on the youth guns really helps handling.
Hopefully my ramblings help.