There's value in ensuring a child has a rifle they enjoy shooting regardless of target. Popping primers can be fun and should be fun - builds good habits.
A youth deer rifle need not be painful but some of the cartridges listed seem to have suddenly become "hard kickers". Proper mounting, form and fit alleviates a lot of that.
My son's progression was basically BB - 22lr - 243 shooting 105's and now he switches between 243 and 30-06. He also grew up shooting 20 and 12 gauge, 2 3/4 shells. It's amazing what a "small statured" person can handle if given proper form and technique along with fit.
Said son never qualified below Expert in the Army and is getting married in 6 months, which means grandchildren soon. I expect them to move along similar path.
Teaching how to shoot is as important as what to shoot. Good rifles in good condition helps make the lessons easier. Never understood the struggle some had. Bullets are the truth. In a rifle that shoots POI to POA - if the bullet didn't hit - shooter error. IOW - don't break the shot till sights are where they're supposed to be - hitting targets isn't that hard once you make sure you're doing that.
