I have raised two daughters as deer hunters. Throw in a nephew and a few neighbors' kids and I can tell you that there is no single best youth rifle. A lot depends on the age, size and maturity level of the child. Sometimes the percussion and noise are more of a problem for the youngsters than the recoil. I suggest that you get good ear protection and that you refer to recoil as "push" instead of "kick". Think about it. Most folks do not associate pain with a push.
That being said, one daughter loved a mini-mauser in 7.62x39 and a Model 7 in .260 (she was bigger)and the other prefered a low pushing 45LC single shot. I bought the H&R Buffalo in 45LC, removed the stock and replaced it with an AR style, adustable stock rather than cutting the wood. She says its a "bad" looking gun. The nephew shot a .223 H&R at first and then graduated to a Model 7 in 7mm-08. Even if you do not reload, several companies now load reduced push rounds in .308, 7-08 and in another kid-friendly round, the .260. But, between the two calibers you listed, I would go with the .223. First, with modern ammo, it is deadly out to 100-150 yards. Morevover, it will be good for paper, coyotes and chucks in later years. I doubt the .357 in a rifle will be used as frequently in the future. Also, the two articles/reviews on the 77/357 I have read have not been overly kind as to the guns accuracy but gave it favorable reviews for size, weight and packability.
You know your child, so pick the one you feel best suits your needs. Good luck and enjoy the time with your kids. Blink and they are grown and gone.
P.S. I have a like new, factory .357 Max rifle that is scoped. It is a shooter and has plenty umph for deer out to 200.
