Having killed a couple of deer with a .30 caliber carbine when I was a teenager I would rate it as one of the worst deer rounds that is legal in most places. I also had the added experience of trailing not just the two deer I killed with it but a few that my cousins killed with it as well. There are few, maybe no, other center fire rifle cartridges out there with a .22 or bigger caliber bullet that I would pass over to choose the .30 caliber carbine. It sucks that bad.
I have never killed a deer with a .357 mag so perhaps it's no better but at least it starts out as a bigger caliber which should make a little bigger hole. And there is a much wider selection of bullets for the .357. The deer I shot with a carbine had a .30 caliber hole in and a .30 caliber hole out. It just didn't produce enough velocity to expand the lead tip bullets we were using back then. That being the case almost all bleeding was internal so blood trails sucked and were hard to follow. This was important because due to it's complete lack of knock down power all deer we shot with it ran and ran and ran. We had to get the dogs to find some of them.
If it was all I had to feed my family then I'd use it and limit my shots to pistol distances. In fact despite my typical dislike of them I'd probably take close range head shots with a .30 caliber carbine if forced to use one again.
Any of the other calibers the original poster listed are 10 to 20 times a better deer cartridge.