Instead of doing a 223 on a DTA which can be problematic (per those who I know who have done it), why not just set up a DMR type AR if you don't already have one? That get's you your 223 option plus another platform for other uses as well. I know some will say "well then you can't train on the same platform".... Bullshit, if you can't be proficient with multiple platforms then it's because you're not shooting multiple platforms enough and all the more reason to. Weapon manipulation training can be done in your house without firing a single round, outside of that its all in how the bullet flies and a 223 isn't going to be even remotely similar to the others so I see it as a moot point for weapon familiarity. That goes completely out the window when you're talking about running multiple calibers from the same platform. As for kids shooting it, it's going to be a heavy rifle and accompany it with a brake or suppressor then any standard (read non magnum) cartridge is going to pretty much be a pussy cat.
I'd throw the 223 out of the mix for the reasons above. After that my 3 caliber setup for that rifle would be:
- 6mm (take your pick after considering all options). Many things to consider however and probably the most important is do you load? If not, 6 creed and call it a day. If you do load, the creedmoor is still a top choice but there's other great choices too. There's way too many factors to just say "this is the end all be all 6mm cartridge". What's brass quality and availability? How much brass prep (fire forming, cleaning up non premium brass) can you justify doing or are you willing to do? What's brass life? What's barrel life? Will I have problems with piercing primers or utilizing the primers in my rifle (small rifle primers)? Is the cartridge optimal for the barrel length and powder choice I want to run. There's a lot to consider.
- 308. Everyone should have one. It's great cheap practice and if you don't load you can buy factory federal gold medal right now for less than 223 match ammo. If you do load you can still load very economically. The difference is that you can load or shoot ammo that is every bit as challenging in the wind as 223 is, or you can take advantage of some of the great bullet offerings and get great performance. Unlike 223, when you miss at distance you'll actually be able to see where you missed. 308 is also a great hunting round if that's a consideration for you with factory hunting ammo on every shelf. You'll just never be a cool kid when you're shooting a 308.
- A 7mm, .30, or .338 magnum. If you already have other options that are good for 1200 yards or so, consider something that is efficient considerably further ideally to a mile or beyond. Something burning no less than 70gr of powder and sending a long pokey bullet out at a mild to wild velocity. Again it comes down to a lot of factors. I've run the short magnums, super magnums, and wildcats and I always come back to 7mm rem mag, 300 win mag, or 338 lapua. They all perform very well, there's a ton of brass out there (338L is pricey but lasts a long time so it's basically a wash), load data galore, and they're all excellent choices for large game. They can also get pretty decent barrel life compared some more hot rod options but if you dance on the edge of pressure, run a really fast powder, or burn 20 rounds as fast as you can crank the bolt you will torch it just as fast.
Regardless of what you choose, remember that you have a modular platform that you can simply order barrels for so changing your mind after the fact isn't a huge issue. As long as you don't go with anything really unusual you should have no problem finding a buyer for a barrel, dies, brass, and bullets to recoup a good chunk of change.