SlimG2, welcome to the board. You will probably get a lot of opinions regarding those cartridges, but they should all do the trick.
The first rifle I bought was a Ruger 77MKII in .300 Win Mag about 12 years ago. I made some mods to it after catching the scope several times in the forehead and I love that rifle. I have used it for almost all my deer hunting over the last 12 years where a rifle is allowed, even though it might be a little more than what I need for deer. When I initially bought, I figured it would be good for anything that I wanted to hunt in North America. Just haven't had the chance to go out of state on a big game hunt. However, I have come to the conclusion that if I were to ever go on a grizzly hunt, it would be with a .375 h&h at the minimum because I do not want to become dinner. Otherwise, the .300 Win Mag is just fine for elk, black bear, etc.
All the cartridges you are looking at seem fine for deer and elk hunting, with the .270 possibly being on the light side. That comment should get the pot stirred up.
I was just looking at the ballistics between a .300 RSAUM, .300 WSM, and the .300 Win Mag because I am contemplating an AR-10 purchase in one of the first two cartridges. They are almost identical to the .300 Win Mag.
At the end of the day, I think any of those cartridges, with the proper bullet selection and bullet placement, will do just fine on deer and elk. If you start hunting smaller game such as antelope, coyotes, etc, you can pick up a smaller caliber at that point.
Personally, I prefer the bird hunting, specifically waterfowling, to deer hunting, but that is just me.
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