Of the two choices listed by the OP I'd go 22-250 for a young shooter. I've got a Ruger compact 223 in my safe for my nephew when he's ready. I haven't killed a deer with my 22-250 but I have with my 220 Swift. 175yds and she didn't take a step. These small southern whitetails aren't armor plated.

I rarely ever take anything larger than 25 caliber when I go after deer.

Those who must raise their voice to get their point across are generally not intelligent enough to do so in any other way.