In New Mexico, you'll be far better off with a .243. Keep the .22-250 for ong range jack rabbits or coyotes. I wouldn't trust the .22-250 to perform on a 350 lb mule deer across a canyon. You need much more gun for big mulies and longer ranges. And, that ain't JMHO, that's just plain fact.
In Texas on small Texas whitetail, .22-250 has a following, but I've really never understood why even in Texas. Sure, it'll work on Texas deer. I figure if a youngster is old enough to deer hunt, he should have no problem handling a .243. My first was at age 11 with a .257 Roberts, personally. If the kid is scared of the gun, just wait a few years. As an adult, there is NO advantage in a .22 over a 6mm at all. Neither have any great recoil and the 6mm is a far better choice. In New Mexico, you can get longish shots and you're talking about deer 3 to 4 times heavier than a Texas whitetail. I haven't looked at the regs out there. Hell, the .22 might not even be legal for deer out there.
The one mulie I've taken in New Mexico was at about 350 yards across a steep canyon in the Guadalupe mountains in SE New Mexico. I took it with a 7mm Remington Magnum.