A true "Mountain Rifle" is one that weighs no more than 7.5 lbs all up and ready to hunt with and closer to 7lbs or less is better. This includes the scope, mounts, full magazine, and your sling. Caliber is not important, any flat shooting chambering from 6mm up to the 300 magnums is fine for most game. Since light weight is the primary goal of a mountain rifle most people tend to work with chamberings 30-06 and down. The 270 you have is as good as any and better than most. The question is, which model 700 do you have. Some can make the weight limit, most cannot. At least not without a stock replacement and other modifications.
You don't have to spend a fortune to get weight down. Simple things like choosing a short action chambering, lighter scopes, mounts, and slings can make a difference of 2-3 lbs with the same rifle.
270 vs 300 mag. Both have near identical trajectories. With premium bullets in the 270 it will give you virtually identical results when game is hit on all North American game short of a 1400 lb coastal brown bear. If deer are the largest animal hunted a 243 is more than enough. Nothing wrong with a 300 mag either if that is what you want to use.
Flat shooting rounds work just as well at 50 yards or less as they do at much longer ranges. This is where a variable powered scope comes in handy. Set at 3X or less is about perfect for close shots. 6X or greater will be all the magnification you need for 400+ yard shots.