So, lots of guys giving a quick answer with little detail, and/or no real reason other than "I have one and I like it" type answers.
I'll give the "why" first, then the what" of my answer.
What does a "do it all western rifle" need to do? For my money, that rifle needs to put a 2"+ wide wound channel 18-20" into an animal. That kills any deer, pronghorn, or elk from pretty much any angle barring going straight away/super hard quartering away going through the stomach/guts first. It needs to be a rifle you can shoot a lot (we all have a budget for recoil, and a budget for ammo cost. Some are larger than others, but they exist for all of us). Your limiting factor in the mountains will be you, or maybe your bullet choice, not lack of horsepower in your chosen cartridge.
You said you want to be able to do that at 500ish yards. You can do that easily with a 6mm ARC. A 108 ELDM launched at 2600fps hits the lower end of its velocity range where it upsets at 600-700 yards (depending on elevation/temp/where exactly you draw that lower threshold velocity line) I killed a mule deer this year with a 108 ELDM that impacted at that kind of velocity, and I would have zero hesitation putting that bullet in an elk at that velocity.
You need to be able to practice with it. That means you need to be able to spot your shots and do a lot of positional shooting. As a bow guy, think in terms of what it takes to be proficient at 50 yards, and you'll have a sense of what it takes in terms of focused practice/reps to be a competent 500 yard rifle shooter in field conditions. Off the bench at the flat range is a very different thing from in the hills or mountains from an improvised position in the wind under time pressure. That practice is a lot easier to make happen with a rifle that costs $1.50 per shot vs $4 per shot.
I'm not saying that something like a 7PRC has no merit for anyone. I would say that the benefits a 7PRC brings to the table don't outweigh the disadvantages until you get way way out there. To maintain proficiency needed to be able to shoot animals at those distances, a guy needs to be shooting several thousand rounds a year *in the terrain they hunt in*, not just at a flat range. If the rifle budget is constrained as described, it's safe to assume you don't have a $5k per year ammo budget. Me neither!
If you had a $1k per year ammo budget, you will be FAR more deadly at 400-500 yards with a .223/77TMK or a 6mmARC/108ELDM and 700-1500 rounds of practice than a 7mm PRC and 250 rounds of practice per year. Even if you did triple your budget so you could shoot the magnum as much, that practice isn't nearly as productive if you're not spotting shots through the scope. Also, hit rates go down as recoil goes up, period. Anyone who tells you their precision (especially from field positions) is just as good with a heavy recoiling rifle as a light recoiling one, hasn't tracked their hit rates comparing the two.
All that said, Tikka .223 is my first recommendation (though some would feel it's at or a little past its outer limit at your 500 yard stated goal distance). Second recommendation is for me a tossup between a Howa 6mmARC and a Tikka .243win. You will get cheaper, better practice with the same ammo you hunt with if you go ARC, but you'll get a better functioning rifle with a Tikka. .243 is hard to optimize for both practice and hunting unless you are hand loading; the bullets that offer a performance advantage over 6mmARC are not readily available in cheap factory ammo, so you don't get as much of the cheap practice benefit going that route. I'd avoid the savages and Ruger American, I've owned both and prefer the Howa mini over both.
If all that is unconvincing, 6.5cm Tikka is the answer. Easy and cheap ammo availability, still on the lower end of the recoil scale in the big picture, and terminal performance to nearly double your stated goal range.
. @Marbles nailed it. Lived and hunted in Idaho my whole life. Everything from moose to antelope including varmints. That 243 will do the job. No need for a magnum. I suggest using that cash to get very confident In your current rifle especially if your planning shots at a decent range. Let me tell you the weather/wind out west can be unpredictable and shooting 600 yards is not the same as a flat range. 
