I think recognize this canyon overlook. This looks a lot like the Deep Creek Canyon in the Flat Tops Area outside of Glenwood Springs CO. If it is there is some really good trout fishing in that creek and some good bear hunting in it as well. Not to mention elk. But it is a rugged place to get in and out of.

you got it Ive hunted that canyon and the surrounding canyons, like deep creek, in all directions for 25 miles or so, eagle, gypsum, rifle,meeker, areas,, etc. for decades

I used to live on a horse ranch below Mt. Sopris in Glenwood Springs so that whole area was pretty much my stomping grounds. Good fishing and hunting in that whole area.

each of our personal experiences accumulate over time and each while probably significant different has a profound effect on what we personally see as the ideal equipment list , and colors our choices.
I prefer to hunt out of a back pack, think mountain man, tactics with very limited minimal but better more modern equipment choices
most of my fellow elk hunters in my group think I'm at least a marginal masochist, because I will almost always, leave camp and am perfectly comfortable under conditions many feel are not tolerable.
personally I want to cover ground , but leave no trace I was there, I select a light flexible but totally durable back pack, minimal equipment,
a fairly powerful rifle like a 375 H&H or 340 WBY a 22lr revolver,a kukri, and a -10F rated sleeping bag , and a parka and poncho, a couple 2 ltr bottle canteens a few tools, mres, and think little of spending 3-4 days away from camp.
if your not afraid of getting lost,, not all that concerned with having more than one meal a day, not worried about spending nights sleeping sitting against , or laying under, a large over hanging conifer, tree,
even if the temps drop and your waking up under a foot of fresh snow, miles from your truck, I can totally enjoy the hunt..
too many guys are convinced they must be back in camp at dark, too many guys in my opinion think they must have a hot meal and a camp fire.
I cover a good deal of carefully pre-selected, terrain, but I don,t travel fast, Im just persistent, I really don,t care where I am at dawn or dusk, I just want to find a decent elk,
most of the guys in our club want to sit around a camp fire and drink beer at night back at the truck.
I've never been all that concerned with huge antlers but yeah Id prefer to shoot something impressive just like most of you.
this style of hunt is obviously not that popular as only a few of they guys Ive hunted with wanted to repeat the hunts even when I'm successful most years, AT least 50% of the trips in at least getting a decent mule deer or elk.

The simple answer, I carry and use what I'm 100% sure works,and what I personally feel is the best choice, for me.
I would Not be overly concerned with the choice in rifles or caliber,
my late hunting partner used nothing but a 358 BLR loaded with 250 grain speer bullets
and my new hunting partner,is absolutely convinced that anything less than a 270 win loaded with 150 grain boat tail bullets will bounce off anything with fur.
the truth is that anything Ive seen used from a 257 roberts to a 458 win, works,
if you can place your shots accurately. (that does not mean a larger caliber won,t hit harder ,but dead, is dead and a 257 roberts 100 grain bullet placed in the heart/lungs is leathal)
your issue is NOT power its confidence in your rifle and Your ability to use it!
personally I've come to like the 340 WBY and 375 H&H, everything I've ever shot, with either rifle drops fairly close to the point of bullet impact,
but I can say the same for Rons 358 win and Bills 270 win
the main issue you face is finding a viable legal elk thats well inside your comfortable rifle range and placing your shot well,
thus good clear binoculars, the ability to read and accurately interpret topo maps, and good boots and the ability to stay warm & dry and in the field from before dawn to after dark,
to maximize,,your chances of being in the right place at the correct time is more critical.
Id bet theres thousands of Elk that are still wondering colorado each year simply because, many guys hunt hard the first two too three days, get frustrated & exhausted,
and spend the rest of the week or so sitting around camp or driving around looking for elk out of the trucks window rather than,
checking out the small side canyons, that might require a hour or so walking to gain access.
PERSISTENCE MATTERS, YOUR PHYSICAL CONDITION, and YOUR ABILITY TO SHOOT ACCURATELY MATTERS
BEING in the correct location at the correct time is not 100% due to luck! you can swing the odds heavily in your favor,
simply by constantly being in a reasonably likely location during the vast majority of the season rather than cooking hot dogs and drinking beer in camp.