I had my first foray into carbon barrels and custom rifles this past fall. As a dyed in the wool Fudd, you should all be proud of me and my willingness to experiment. I handled three and shot game with two so I feel that I have a certain level of expertise that should allow me to comment on such things.
There are a lot of circumstances where this is the way to go. Chief among them, in my opinion as an avid Nova Scotian hunter, is western big game. The two I shot game with were a custom (unknown builder and can't remember the action right now, obviously because it wasn't a Pre64 Model 70 so how important could it be), and the second a Christensen Arms. Both 28 Nosler, so not a cartridge that wants a short barrel but an awful lot of a good thing. My cousin also has a Christensen in The Lord's Caliber - 300WM, which I saw used to predictably good effect. If I was a guy that hunted out West and was inclined to wander into the hills, I would definitely be looking at a 300 with a carbon barrel. Whether the 300 was the long or short Winchester, I don't think that it matters. The weight savings with the carbon barrel and the transfer of the weight well back between your hands is a different, but very pleasant experience. My cousin is a fan of high powered scopes, which I generally am not. But I have to say that, due to the light weight of the rifle, you can throw on something somewhat silly (like the Mark 5 5-25x that he has on one rifle, or the Swaro Z5 3.5-18x on the other) but not suffer with an overall weight penalty. The rifles themselves are, as you would expect, incredibly accurate. I wondered about the weight between the hands and the effect on my shooting. The custom fit me better and I made three 330 yard hits on a big whitetail buck from the strong-side standing position. The Christensen, I found, was low in the comb for the height of the scope. That being said, I made a fast 300 yard shot on my mulie buck from a quickly adopted kneeling position which sent him down the hill at more than a medium pace.
So, long credential-justifying story short, if I was out west and wanted a serious hunting rifle for those sorts of excursions, I wouldn't hesitate to go with a custom job (either a factory custom like Christensen or full build like ATRS) on a 700-footprint custom action and a 1-10" twist carbon barrel with your choice of "medium 300" reamer run in the chamber end (this is an important detail. Do not chamber the muzzle end...it ends badly). Add a high quality stock (Wildcat for even more weight savings) and a scope like the Leupold 5-25x and you'd be well equipped for anything you could be wanting a moderate-magnum 300 to do.