How light do you consider "light"? In my book, anything much over six pounds is a certainly a lighter rifle but not really a light mountain rifle. As mentioned before, the Kimber Montana is a factory rifle for around $1000. I got one in as my mountain rifle in 7-08 and it weighs just a hair over six pounds with a scope, sling, and four cartridges sporting 140 grain bullets in the magazine. It shoots MOA with 140gr TSX's leaving the 22" barrel at 3000fps, which has proven to be plenty of medicine. Last fall it even proved itself on a large moose about 6 miles back in the mountains. The Kimber Montana is a good choice if you want a mountain rifle for a good price. It's hard to build a 6 pound gun for less. However, the accuracy of the Kimbers seems to be variable (mine shoots only one bullet at MOA, and I tried a LOT before I found it!), so if you have the time and money and want a special rifle for the mountains for under $2000, you'll have to build it.
This is what I have in mind:
I chose the 7mm bore on my Kmber since it was possible that I would end up shooting a moose with it, but if I were to build a custom rifle for only mountain deer, goats, sheep, and maybe the occasional elk I would probably go with a 6.5mm in a Savage short action. The reason for this is that Savage short actions will fit cartridges close to 3" long, whereas my Kimber takes nothing a hair over 2.8". That extra .2" comes in really handy when it comes to powder capacity and seating the long, high BC 6.5mm bullets. In short, you can't do without it if you want to use high BC bullets. I have a Savage short action target/tactical rifle chambered in 260 Rem, and it spits the 130 grain Bergers and Accubonds at 3100fps using RL-17 out of a 25" barrel. In a mountain rifle I would go with no more than a 22" tube, but even so I'm guessing you could still get between 2900 and 3000 fps from a 130 grain pill, with a much higher BC than anything you can fit into a .30 caliber or 7mm short action.
The Savage action will certainly weigh more than the small, light Kimber action, but it will only cost you less than $400 to pick one up new (less for a used one), and you could put the money you save toward lightening the action (fluting the bolt, skeletonizing the bolt handle, etc.). I think the Savage action will be the more accurate of the two as well. I'm guessing (and I intend to find out someday!) that with a custom 22" barrel in an ultralight contour, a Savage stainless short action, and a Lone Wolf Summit XL stock (under 15 ounces) you could end up with a gun very near 6 pounds that be a dream to pack and will shoot .5xx-.6xx BC bullets in the 130-140 grain range at 2800-3000 fps. Wind drift is minimized, sectional density is phenomenal, and down range energy is plenty for medium sized (ie mountain) critters out to 500 yards.
I would reject 200+ grain .300 caliber bullets out of a short action. Bullets that long will seriously cut into powder capacity in a .308 case. My personal preference is also to stay away from the WSMs, since the extra powder capacity they offer comes at the expense of requiring a 24" barrel, which is kind of going backward in the weight category. Even if you were able to match the ballistics of a 260 Rem (or 260 Ackley) with a .30 caliber bullet, recoil in a 6 pound rifle shooting it would be frightful!
I put the 3x9 Leupold Ultralight (vx-II) on my Kimber, and it's what I would probably go with if (when??) I build my 6.5mm mountain rifle. Sure, a bigger, higher power scope can add an advantage beyond 500 yards, but it's just not worth it to me. It's also a neat idea to use a 20x scope as a spotter in the mountains, but when I'm trying to decide whether that ram 1 mile up the mountainside is worth going up after or not, I'm still going to want a 40+ power spotting scope to help me decide. In my mind you can add a lot to your rifle to make it do everything, but the more you do that the further you get away from its main purpose: to be very light and easy to pack in the mountains.