Welcome to LRH!
One of the reasons you have heard so much different opinions is that there are many cartridges that can get the job done.
Add to that some people believe yo need the best and biggest Super-Wiz-Bang to shoot game at 400 yards.
Others will have one cartridge that they believe is best for everything.
Reloading would be beneficial for you.
I like the 7mm cartridges for the bullet options and better BC (Aerodynamics) of the bullets. Nothing wrong with the 30 caliber, but you can do as much ballisticallly with less bullet weight, therefore less recoil. The 270 will also work for you as well. Again, not as much bullet selection for LR shooting/hunting as the 7mm or the 30 caliber.
The 7mm-08 would do all you wanted it to do.
I would prefer the 280 Remington, and there is sure nothing wrong with the 7mm Rem Mag.
There are many different ways to skin the cat.
So please do not expect to hear a resounding one cartridge come to the top for what you are asking.
The other limiting factor is being stuck with factory ammo. Both because of its cost, limited bullet selection or whether your barrel likes that particular load.
Depending on the stock you get, weight, and your own sensitivity to recoil when shooting from field positions (stay off the bench as much as possible). A muzzle brake
may be a good thing to consider. It takes the recoil out of the picture and allows you to spot your own shots.
A 4.5-14 is more than enough for big game out to 1,000 yards. I used a 3.5-15 this year and took a antelope buck at 1,037 yards. I did not feel handicapped with 15x in any way. It was a 7mm and it had the performance level of the 280 Remington. I like the Huskemaw
, VX-3's, Mark 4's, and the NightForce (have all of these scopes).
Since you are considering elk and other mountain game, be careful about getting a real heavy rifle. Once you get up in elevation and steep terrain you can come to hate a heavy rifle, that you once loved on the plains shooting from the bench a few feet from your vehicle.
If you want my opinions on reticles, brakes or anything I else I would be glad to respond. There is a lot of good folks with good info on this site
When it is said and done, buy what you want. It is your money, and if you do not like or feel comfortable with what you get, you will never be happy with it.
Good luck in your endeavors, and practice, practice, practice from field positions.