Both my boys killed their first deer with a .20 ga, which kicks more than a .243. My oldest son killed his second deer with a .243 at age 10 and he's been using it ever since. After seeing the .243's results on deer, I bought my own .243 for deer / coyote and leave the .300 WSM for truly big game.
The .243 is perfect for a 10 year old IMO