Everyone has their own ideas about things, nothing wrong with that. For what it's worth, I'll join in with my thoughts.
Your list of animals goes from small, rather easy to kill antelope at most likely extended ranges, to deer which can range anywhere from small to very large, and then on up to elk. That's a pretty tall order (in my mind at least) for any of the cartridges you listed to do a good job.
I've never hunted elk, would like to though. However, the late Bob Milek wrote an article one time pertaining to the 25-06 as an elk cartridge. He gave it pretty high marks. Bob was known as both a great guy and good marksman, one who knew quite a lot about rifles and loads, and put many of them into action on game. To my way of thinking, Bob was the kind of guy who had enough experience to make the 25-06 work on elk. I had the great priviledge of talking to Bob one time. He was indeed a very good man. I always enjoyed his articles and learned a lot from him.
The 25-06 is a great round. I've taken five antleope with that cartridge, a mule deer doe and a whitetail, plus a Corsican Ram. It's a great cartridge. For me personally, however, I consider it a middle of the road round when hunting larger game that antleope / deer.
The 7mm-08 is an inheritantly accurate cartridge, capable of shooting heavier bullets than the 25-06. It is generally considered a mid-range cartridge. It would be a great deer getter at moderate ranges, but could perhaps be a bit lacking on long-range antelope. My son took a huge Wyoming mule deer buck with his 7mm-08, at about 150 yards.
The .308 is an old standby, a good one too, and sadly one I have never owned.
You mentioned three of the "most interested in" cartridges, but you haven't necessairly ruled out other rounds, and are open to the idea of one rifle for everything.
My thoughts are go heavier since elk are involved and possibly heavy deer. A good 7mm magnum would work well for all the game you mentioned if considering just one cartridge. It's relatively flat shooting, packs a punch, and has a variety of bullet weights to do the job well on all three species.
There are so many good calibers available that it's just hard to choose only one. That's why we all like variety, and strive to lay our hands on something "just right" for the task at hand. It doesn't mean that we can't use something different, either lighter or heavier, it just means we look for the ideal cartridge, indeed if one exists.
For me, and elk hunting, I would reach for my 7mm STW or a .338 mag., or even the .300 win mag. simply because I have them in my gun rack and can lay a hand on either one of them. But, if all I had was a 25-06 and a burning desire to hunt elk, I would do it, but would choose my bullet carefully, notwithstanding distance and shot placement.
My two-cents worth may not be worth much, so take it with a grain of salt.
