I have owned and used my old SAKO .375 for many years and like it very much, but the 9.3x62 is a very fine cartridge and with top loads/bullets in each I don't think you would see much difference in performance in the field.

From a recoil standpoint, sounds like you are avoiding a muzzle brake. If you added a brake, either would be easy to shoot from a shoulder standpoint, not so easy from an ears standpoint. Without a brake, on the surface of it, the 9.3 might be attractive in regards to lighter recoil, but I am not so sure about that. You have to choose the rifle carefully. In general, I am guessing that 9.3's will be somewhat lighter {most I have handled were} than many .375's and that difference may make for tougher shooting from the 9.3, tho in theory it develops less recoil. A good example is some Husqvarna 9.3x62's that weigh in the 6.5 lb range. You would know the difference between those and a 10 lb .375 for sure. I had two 8x57 Husky's that developed felt recoil that was much more severe than that generated by my .375. Stock design had a lot to do with it, also. Anyway, I suspect that were those two guns rebarelled to x62 and shot with heavy loads, the recoil would have just plain nasty.

Both are great cartridges for all North American game. Recoil tolerance and just plain subjective "likes" might be the deciding factors for you.