I've killed one elk, a Roosevelt bull. I used a 338 with a 4x Leupold and 225-grain Nosler Partitions handloaded to about 2,750 fps. Next time I'll use a little more scope and a little less rifle.
I'd recommend that you start with the elk, not the rifle. You mentioned western states, so that's a Rocky Mountain elk and not the larger Roosevelt elk.
Go to Cabelas and compare full-body mounts for deer and elk. Elk aren't that much bigger than deer, but their bodies are considerably longer. Look at them from multiple angles and visualize the worst shot you might get. You'll quickly see the importance of penetration. To me, that says you want a premium bullet like the Nosler Partition, Barnes TTSX, etc.
From there, you have to get the bullet into the vital zone, which is about the size of a toilet seat. Determine your maximum range, decide whether you plan to dial or hold over, then look for a scope that lets you go that route. Elk are diurnal so you don't need a giant objective bell, which helps keep the weight down.
Then look for a cartridge that a) has multiple factory loads with premium bullets, b) shoots flat enough to work with your method of getting hits and your max range, and c) is widely available these days. This will limit your choices more than anything. The 308 will do if you have enough experience hunting elk to pass up an iffy shot. For a first-time elk hunter, I'd start with the 30-06 or 300 WSM. The 300 and 338 WM are classics, but they have the drawbacks that others have mentioned. The 35 Whelen is good and factory ammo has improved a lot recently, but full-power loads kick like a mule in standard-weight rifles like the Remington 700. Learning to handload is a rabbit hole but it opens up a lot of options.
Then choose a rifle that's chambered for that cartridge. Not every premium load will shoot well in your rifle, so buy 2-3 boxes of each and test them out to 3-400 yards. Remember that an elk's vital zone is huge compared to that of a deer so you don't need match accuracy. 1.5 MOA is plenty IF the bullet will hold together and penetrate 4-5 feet of elk on a raking shot. Pre-COVID, the ammo tests might have set you back $400 but I'd plan to spend double that if not more now.
Then you'll have a pretty good choice for elk.
For the rifle itself, I once saw a pretty good set of criteria:
- Rifle with scope and sling must be under 8lbs
- At least .30 caliber
- No porting, brakes, or other superfluous hardware
- Non-glare stainless steel in a good pillar bedded synthetic stock
- 22" barrels—except in the magnums that need a 24" barrel for velocity
- Simple trigger group, bolt and safety designs
- Shoot flat enough that 2" high at 100 ensures that holding high hair at 300 yds will put it in the vitals
- Iron back up sights
- Scope with an objective no larger than 40mms held with field detachable mounts (scabbard friendly)
- Shoot premium bullets under 1.5" throughout the entire temperature range that can be encountered
The original list didn't mention that the rifle should stay zeroed wet or dry, hot or cold, but I think that's understood.
Good luck. Elk hunting gets in your blood.
Okie John