I have two daughters. I have several guns that they use without complaint. I do not say "kick" around them. I use "push". The .243 is not bad at all. Been a kid's first gun for decades. You can also use the reduced recoil loads that Remington and others are using. They are available in 7-08, 270 260, etc. but you have to be sure the gun's stock fits the kid. If you opt for reduced recoil, they also come in 30-30 and there is not much push there at all. Marlin makes a 30-30 for kids (Marlin 336Y). Otherwise, we have used 243, 260, 6.8 spc, 45 Colt, 30-30, 300 Blackout, 7.62x39 and 7-08. I think a Remnigton Model 7 in 260 or 7-08 is the cat's meow for kids. Ammo for both calibers is available in reduced recoil, and your kids will not be undergunned for as the grow up. In fact, I will slip off with a Model 7 when they aren't going. Lite push and easy to manuever in a box stand. My two cent opinion on .223 is that, with the new ammo, they will take a deer. But they are best reserved for the experienced hunter who has had neck or shoulder surgery just weeks before the season. There is little margin for error in shot placement. Those errors will happen more often with kids regardless of how much you practice. The low push options are just too many as mentioned above to go with a marginal gun. You will never be accused of spending too much time with your kids and hunting is a good way to do it. My wife thinks our daughters come to me to talk about things to her exclusion. Its the hours and hours and hours sitting in my lap and next to me in a box stand that did it. Good luck.
I'm a huge advocate for 7mm-08 for anyone. From youth to adult it's a great round for hunting whitetail in my opinion.
I have settled on a Savage Axis youth model in 7mm-08 with a Witt Machine muzzle brake added on.