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.
People are like rifles. Some are tried and true, having great eyes, personality, and fun to be with. Others never seem to hit the mark with you. Still others go off half-cocked. Still, it's nice to know most of them.