I'm unwilling to start kids with semi autos.
I've started my 9 year old granddaughter with a single shot bolt action Winchester M60 from the '30's.
She loves loves it.
I would no more trust a 10 year old with a semi auto than I would a drunk with any gun. Gun handling includes carrying a gun for hours through the woods, not just shooting at the range.
Sorry for being blunt, but it's important that kids learn safe gun handling.
Posting videos of kids shooting handguns or semi autos is unconvinvincing.
The videos capture a few moments. Not the routine gun handling of the subject.
I've taught many kids about safe gun handling.
No disrespect to the kids: just a realistic view of their ability to safely handle a gun.
I think it really depends on whether there is close supervision by an adult who will instill discipline and respect for firearms.
I started my kids with a bolt action single shot Marlin.
On the other hand, both started with a gas operated 1100 and 1187 20ga shotguns for recoil reduction. Only loaded one shell at a time at the trap club. In the field they loaded more but were under close supervision.
Regardless, instead of a 10-22, I'd suggest something like a Savage Mark II FVT. The F stands for synthetic stock. The V is Varmint (bull) barrel, and the T stands for target model. At some point, upgrade to a wood stock, like a Boyds, and make it a father-son project for final fitting, maybe glass bedding, sanding, finishing and sealing.
This gun is more expensive than a 10-22, but that's due to the receiver and barrel being drilled and tapped for the Williams target sights. The target sight from Williams is $99, the globe front sight is $51, and the front and rear sight bases are just under $21 each. So that's $192 total. These are good quality sights. This is an entry level TARGET rifle, not a plinker, though it certainly can be used for putting holes in soup cans.
Note that there are two versions of this gun. The Mark I FVT is a single shot. The Mark II FVT has a detachable five round box magazine. The Mark II FVT would be more useful down the road, as it can certainly be used for hunting or varminting. And the receiver is drilled and tapped for a scope base. The rear target sight is easily removed and a scope can be attached for hunting trips or practice.
BTW, I have a couple of Savage bolt action 17 HMRs, my son has two as well, plus I'm setting up a Mark II as a suppressor host. Been wanting a target sight model, and am considering either an FVT or retrofitting one of the other bull barrel Mark II's with the Williams sights.
Sorry about the lousy image. You can find the gun on the Savage website in the List Firearms section under Mark II.