Ballistically, a "properly loaded", whatever that means, .357 is not equivalent to factory 30-30 ammo. There are two things missing. Bullet weight and/or velocity. I don't believe you're going to get 2000+ fps, across a chronograph, out of a .357 revolver or rifle with 150 or 170 gr. bullets. If it's being done I don't want to be anywhere near. Until the advent of heavy, 300 gr., 44 cal. bullets the 44 mag lagged considerably behind the often and wrongly maligned "thutty-thutty". Check the ballistic tables in any loading manual or run a load across your own chronograph and make your own calculations. There's this thing called "powder capacity". Comparatively the 357 is significantly behind the 30-30.
Regarding killing deer with a 44, C&B revolver, sure, it can be done and has been. The favorite tool of deer poachers is a 22 LR and the 44 carries more energy than a 22. Not a lot but still, more. That isn't to say it's the best tool for the job, which it isn't. Having said that, a careful, experienced hunter, as you say you are, should be more than capable of taking a deer with said revolver. I almost took a shot at a skinhead whitetail last year with a 45 cal. flintlock pistol loaded with 40 grs. of Fffg, and would have, had the deer presented a shot I was comfortable with. There is a little more energy and velocity in my flintlock pistol than any of the revolvers, Walker included. Ballistically it has to do with barrel length. It's still very marginal. The deer didn't present the shot and I didn't shoot. It is imperative you maintain that kind of control, and, from what you've said, you are capable. Would I do it..........maybe.....maybe not. Conditions would prevail. Ordinarily I carry a pistol in the same caliber as my rifle during muzzleloading season so it's unlikely the opportunity will present itself.
Obviously local laws will determine the legality of using a C&B revolver.
Vic