I agree that a .243, as well as the 6.5 CM are excellent for the type of hunting you are describing. I picked up a 6.5 CM Tikka Superlight stainless synthetic to be my sheep/goat/ibex/mouflon/etc specific gun. could i take my 30/06, or my 338 WM? yes i definitely could. but i didnt have anything in my gun collection between .223 and 30/06 so the Tilka fills that gap. lightweight, flat shooting, and hits hard enough at distance for what i plan on using it for. my humble opinion, gentlemenI'm no expert and will be watching this thread as a sheep and goat gun is in my future as well... But to throw something new and different out here.. I've been drooling over the Kimber Adirondack in a 6.5 Creedmoor... Just wish it had a couple more inches of barrel length. What I really want is for them to wake up and chamber the Mountain Ascent in the Creedmoor, same weight and a 22" barrel vs. the 18" on Adirondack. I'll give em a year and see what comes. I would love to put the sleek and light weight Z3 3-9 x 36 scope on it but for 500 yards I would go with a Z5 3.5-18 or the Zeiss 3-15 that Kimber sells with matching camo. Thinking I'd also drop the bucks for the one piece aluminum Talley rings.
The Creedmoor is a legit 500 yard caliber with recoil akin to a 243. At 500 yards it delivers 1200 Foot Pounds at still over 2000 fps, that should take down a sheep cleanly and not hurt to bad to shoot in the lightest weight production bolt gun available.
But I have never hunted sheep nor high mountains... I just know that at altitude those ounces can feel like pounds so I want a light weight gun that can do the job. I'm sure there are many other great options with marginally more weight and much more power.