I've been hunting for over half a century, for those of you who don't know that was before the .44 mag and all those other new comers Mags were on the market. I either pulled the trigger on or withnessed the .357 mag kill a number of animals (Deer, Black Bear, Hogs, ground hogs, coons and a ton of cans). In the early days we used open sights, so ranges where short (20 to 40 yards). We shot what ever we could buy at the sporting goods store. My Dad and My Uncles all had the same kind of gun, Smith and Wesson, 8+ inch blue .357 mag. I learned to shoot Dad's with .38's. I can't remember ever tracking a deer very far and remember being pulled out of the way by my Dad as my Uncle shot a bear out of a tree. Bear was dead when he hit the ground but, you know how Dad's are.
Fast forward to today. That old Smith and Wesson of my Dad's nows sits in my gun safe, retired. However, I still put the 10 inch barrel on my Contender and deer hunt with it. I hand load four different bullets for the Contender. I use heavy doses of H110, 296 or WC 820 to push a hard cast #357429's, Hornady SSP 180's, Remington 180 JHP's, and Sierra 170's JFP. I have not Chronographed them but, in the Contender I sure I'm getting at least 1400 fps from these loads. I have used shoulder shot with the #357429 and broken the on side shoulder, destroying the lungs and coming to rest just under the skin on the off side (I've done this several time at varying distances up to 75 yards and some times have not found the bullet). I've done the same thing with the Hornady 180's and the Sierra 170's but not as many times. I have only used the Remington 180 JHP on lung shots and until I try it I can not recomment it on the shoulder shots. It's hell on lung shots though. I use a Leupold 2X scope barrel now and I hunt from a number of ladder stands with good shooting rails when I use the .357 out to 75 yards (it will do more I just limit myself because of eyes and age). Speaking for the old girl she has done a good job for me over many years. She's cheap to shoot, does her job and is always ready when I want to go hunting.
Now to answer your question. Yes the .357 mag will kill any whitetailed deer in North American as long as you do your part and put the bullet in the right place. Are there better calibers for hunting deer/hogs/bear? My answer is yes (at least just as good). I use some of them. .357 MAX, .44 mag, 30 30, 7-30 Waters and the .45 long colt. These calibers do not kill the deer any "deader" they do allow the hand gun hunter to make longer shots with a greater margin or error at the expense of more recoil, a louder bang and more expense.