Have you considered air rifles?
Around here, there's just about no ammo available except, as you say, larger game ammo and shotgun.
Not exactly what a youngster would find a lot of fun to shoot.
No pistol, no .22 or .17, either.
And even airgun supplies are not as plentiful as it was.
Lots of folks must have discovered airguns lately.
And not just as an interim solution, either.
They are very accurate, the better ones are powerful enough for small game, and will probably become a permanent addition to your collection.
I prefer the break barrel, spring piston powered versions.
They recoil quite a lot, for realism, and seem to last forever.
$150 - $250 will buy most.
If you're not familiar with airguns, check the video reviews of dozens of them at pyramydair.com