NM resident here too. My boys love shooting my Tikka 223's, but I also have a 243, 6.5 Creed, 22 Creed, and more magnums. I've had them swapped around in Tikka compact stocks, regular stocks, XLR Element chassis with adjustable buttstock, RokStok, Stockys VG. Scoped with everything from cheap Vortex, Leupold VX6HD, Nightforces, Mavens...
I've hemmorrhaged way more money figuring out what works than I'll ever admit to my wife. Here's what I would do:
Scope:
Maven RS 1.2 in MILs. It's a little more money, but a "buy-it-for-life" purchase. You won't regret it. Not too heavy. Just leave it on half mag if you want your kids to focus more on the shot.
Rings: Sportsmatch or UM mediums
Stock: Tikka factory compact stock with Limbsaver recoil pad. Get the vertical grip and if needed install a
kydex cheek riser. You can set up the factory stock the same way if an adult is shooting or use the spacers on the compact stock.
Probably most importantly - get a silencer on it. If budget is a concern I really like the Otter Creek Polonium 556 for my 223's and 243. Great bang for the buck if you can't swing titanium.
My boys have outgrown the compact stock so I'd be open to selling it if you're ever up in ABQ.
FYI - your next build should be a 223. My boys have taken mule deer, elk and oryx with it and the barrel life is nearly infinite...