There's no doubt that it would work (or let's say shoot) effectively - but I would sure not like to tote that rifle around for all purpose hunting of the game that you mentioned. I think the .308 is a great cartridge, versatile and potent enough, and inherently accurate. So let's stick with the .308 per your question. The 26" tube might get you another 80 - 100 fps muzzle velocity, on average, over a shorter sporter type barrel in the ubiquitous 22" range. I don't think that extra 4" of barrel gets you significant extended range in a .308, and I don't think that's a good enough tradeoff for the extra weight and size. I'm 60 though, and objects that I have to carry seem to grow and gain weight every day. Most lighter, hunting weight rifles today shoot more than good enough to hunt with and can be tweaked, if necessary, to fine accuracy. I'm not suggesting an ultra-light, but there is enough middle ground between the ultralights and the varmint/target rifles where you can save a few inches and a couple of pounds. I'm sure many of the folks here could quickly and precisely tell you how much yardage (or PBR) you would gain with the longer tube and different bullet weights, powders, etc. I don't think it is that significant.
I have three Model 77 MkII's - two .308's and a 6.5 Swede. One is an International, One is a stainless/synthetic, and one a blue/walnut. All shoot right at 1" with factory ammo. I have installed Timney triggers in them.
With the variety of game that you mentioned, you are bound to be doing some walking, climbing, and stalking. That's part of hunting, so If you are going to only have one rifle, I think you have to consider that aspect as well. It's amazing how heavy 11+# of rifle, ammunition, scope, and sling can feel like under hunting conditions, especially with a little altitude.
Good shootin' and Good Luck