I think that bullet placement and quality would matter far more than the caliber being used. Just my opinion. Remember the old Swift ad tag line, "Swift bullets, the cheapest, most important part of your hunt." A box of premium bonded or x-bullets is cheap insurance for a successful hunt. A guided Western Elk hunt often costs more than a trip to Africa. My choice would be to use the best bullets I can obtain that shoot well with my rifle at expected shooting distances.
We were just having a discussion on this subject at my gun club among about six very experienced Western hunters about rifles used to kill Elk and Antelope. One hunter used a 270 and was teased by his hunting buddies for being "over-gunned". They were all using 243 for elk. All of them except one tagged an elk with their 243's. Where I live the 270/7mm is the minimum legal caliber for elk but obviously in years past out west they were not being limited or were too ignorant to know. I know that 270 works fine on elk as does 30-06 and I know that often bigger is better up to a point. I would not use a 243 for elk even with a good bonded bullet but I also know that as the ranges increase the advantage offered by the 375 begins to fade. It is a big flat based bullet that has to plow a large hole in the air and it loses alot of its power after 300y. If out west in territory where shots exceed 300y, I think a 300WM or even a 338WM or similar cartridge might still be a better choice for elk or mulies. If you know you will be in timber where the open shots are closer the 375 is a great choice. Plus with the right scope and ballistic knowledge it is accurate and capable of taking elk at about any reasonable hunting range. Again, shot placement has priority. As long as you are capable of making a good shot it will do the job, me thinks.
My long winded point in summary is this: A well placed hit with a 243 at 500y is better than a poor shot and a miss or worse, a wounded animal with the 375 attempted at the same distance. I can hit an elk at 500y with my 375 but the groups that I can achieve at that distance do not give me great confidence that I can consistently place all of my shots inside the vitals at that range assuming a 12" circle. On the other hand with my 243, I can consistently hit a 3" circle at the same distance. It does not insure a clean kill but the accuracy is there. At altitude the air is thinner and the bullets fly longer with less energy loss but at some point the shot is no longer going to be ethical and the hunter should stalk closer or wait for a better opportunity. Your limits will vary as do my own.