Most kids as young as 8 are not ready to hunt deer. In Maine it's not legal for kids to hunt until they're 10.
That said, Federal Premium makes factory ammo for the .223 Rem with a 60gr. Nosler Partition, which can be a fine deer bullet. My Tikka .223 with the standard 1:12 twist stabilizes the 60 grain Nosler Partition quite well, but my grandkids use my two .243s.
I've seen what less than great hits with a .22-250 Rem do on deer and it's often not good. I don't know if there are readily available factory rounds for that caliber that are good for deer hunting. I don't recommend using varmint bullet factory loads for deer.
I used the .22-250 to kill deer in the past, but the 1:14 twist couldn't stabilize 60 grain bullets, so I was handloading 55 grain solid-based bullets that were available back then. It's unclear whether the 1:14 twist would stabilize 55 grain Barnes TSX bullets or Nosler 60 grain Partitions. Mine wouldn't stabilize 60 grain bullets.