Top is my 12 yo sons Howa youth 7mm-08 came with adult stock picked it up for $450 13 rounds through it. They go new around $550
Bottom is my 13 yo daughters Ruger Compact .243 Bought new for $650. Does not shoot factory ammo worth a damn. took me longer to find a handload that did shoot less than 1". If you go with the Compact go with the lightest cal as the small size and weight compound the recoil. It doesn't punch the shoulder as hard as the 7mm-08 but the recoil is more violent, if that makes sense. Even so my 6 yo son still shoots it.
Guns that I've tried that my kids had trouble with that got sold short notice. Rossi matched pair .22 .410 and Remington 770 .243. The $300 770 shot better than my daughters Ruger just didn't fit her as well, didn't like fit & finish or the long action on a .243.