I think the .243 is a fine choice. I'd think 6mmARC would work too but at the end of the day I think 6.5cm's cheap ammo availability is a plus, and recoil doesn't seem to bother my very recoil-averse 11 year old girl, nor my very recoil averse 8 year old boy. If you don't reload, the availability of cheap but decent practice ammo such as what AAC offers, is a huge deal, IMO. I *do* reload so my choosing a 6.5cm for the kids was based more on long barrel life and ease of finding components.
All three of my kids started on .22lr then 5.56 and they've hunted with 5.56 but never actually shot anything with it. Is it a perfectly sufficient hunting caliber? Yes. But it isn't legal everywhere. Step up to 6mm bore size and you're legal everywhere. From there....I think there's a lot of good choices ranging from 6mmARC on the small side to 6.5CM on the large side and that includes the .243 and I wouldn't criticize any of the choices you might make in that range and all of them are more than sufficient with good ammo to the sort of hunting ranges any 11 year old will likely be capable of shooting anytime soon....and then some.
As for rifles....there's a lot I like about the RAR but I am not much of a fan of their stocks or magazines. I'm even less of a fan of the Savage. I'm not saying they aren't great values, just that I personally like better stocks and a bit more refinement than the Savage offers, at least. Between the two I'd get the Ruger, hands down, but I was in OP's exact position recently and I picked up one of the threaded compact Tikkas (not my first tikka but my first T3) in 6.5cm and it has been a great easy-button solution. My oldest is old enough that I *thought* she'd be able to shoot it well with the longer stock spacer. I was wrong; she could shoot it but had a lot of consistency problems with it and those vanished when we shortened it back up to the youth length, and I've already learned long ago that a full sized adult (me) can usually do passably well with a youth stock if they have to, so all three of my kids (plus me) can easily shoot this rifle.
I'd also look at the Sauer that others have mentioned. YMMV, I 100% get the idea of budget constraints but if I have to choose between the Ruger or spending another $200(?) on the Tikka, I'd choose the latter. The Savage wouldn't even be an option IMO.
And, honestly, a significant part of that is because I prefer the available vertical-ish Tikka grip over the Ruger's grip angle. That might seem like a small thing but I believe their vertical grip is much more natural and lends itself to a faster learning curve and comfort from a variety of field positions. YMMV. If the genII Ruger had a better grip angle I'd reconsider.
All three of my kids at 400 yards, prone, a few weeks ago. Still learning with a lot of room for improvement but they're learning.
