It seems like the 308 Winchester has a fixed spot for long-distance work. Why is that? Is it just ultra-accurate?
It's a very accurate cartridge. Arguably more accurate than the .30-06, but only measurable if you're an expert shooter already shooting sub MOA groups. Even then, the difference is small but that difference to a shooter doing .3 MOA groups, in a competition, could be significant. TO a hunter or casual target shooter, it's VERY unlikely you'd be able to tell the difference in accuracy, between the two cartridges, if you could hypothetically have completely identical rifles, with ONLY the cartridge being different. Unfortunately, this is the real world and you'll almost never see those conditions. The point is, the difference in accuracy is relatively small and likely un-noticeable to a majority of shooters.
Doesn't it drop more than say, a 30-06?
Yes, but long range shooters typically don't care much about drop, as long as it's reasonable. All it involves is knowing the distance you're shooting and dialing in the optics for the range. The .30-06 would need to be dialed in as well, the only difference is the .308 need a touch more drop dialed in for a particular range. Not really a game changer. If it was all about flatness more people would be using .300 win mag, .270 Winchester, or 6.5mm magnums. Cartridges with less time of flight will also spend less time in the wind, and therefore, drift less. This usually means more recoil, because more force is needed to push the same amount of mass, at a faster speed, out of the barrel. Accuracy, drop and wind drift, don't matter if you're flinching so much that you can't make accurate shots.
Do I believe that .308 is the ultimate, 'do it all' long range cartridge? No. But it's good at what it does. It's at about the max level of recoil that most shooters are comfortable with, bullets have decent BC's, especially once you get over 180 grains and the cartridge is efficient for what it does. Can't really go wrong with a .308 but IMO there are better choices depending on what/how you're shooting and how much recoil you can handle.
I think one of the biggest factors with long range shooting seems to be recoil. When you're spending the whole day at the range, practicing, you could develop a sore shoulder or a flinch from a high recoil cartridge. Personally, I don't think .308 win or .30-06 are big kickers, but some people will disagree.