If you want to go Classic, a 6.5x55 or 7x57 (275 Rigby) or even a 30-06 with low recoil loads (at least a couple manufacturers make factory downloaded ammo with about 125 grain bullets, very effective on deer). All those you should be able to find ammo, if not, and for a kid, I'll send you a box or two
More modern, the 7mm-08 is a great option. I have a dandy little Ruger Mark II compact/youth gun with fine walnut and blued steel as God intended rifles to be. I also have a 6.5 Creedmoor (Manbun) and love shooting it. In fact it is my truck gun for coyotes or wood chucks, porcupines, skunks and if in season and I need, can pop a deer with it. Look up something like the Hornady factory ammo ballistic chart and compare the 243, 6.5 Creedmoor, and 308. If I remember correctly, the Creedmoor carries twice the foot pounds of energy out to 500 yards as a 243. and does so with close to the same energy as a 308 but with less drop and less windage correction than a 308. And shooting a Creedmoor feels a lot more like a 243 than a 308. That in essence is what all the hoopla is about!
I have the equipment but haven't set it up yet to start handloading. I'm sure the venerable old classic Swede is capable of everything the Creedmoor is. Just perhaps not in as much selection of factory ammo.
I do think with the proper ammo a Creedmoor is capable of taking an elk, especially a cow. The Swede is apparently used on Moose in Scandinavia. I took my first (and second) Creedmoor to Africa. on that first hunt, my PH Mark, was the guy who guided the Hornady people when they first took the Creedmoor out to prove it as a hunting round. He was very impressed, especially with the Interbond 129 grain bullet. I believe they shot a kudu and eland... Although Mark thought it was a bit light for eland.... On a few longer walks looking for whatever Africa offered up, he would say "why don't you bring the Creedmoor". The first such comment, I asked what if we see a good kudu.... Mark shrugged his shoulders and said, "shoot it with the Creedmoor"!