Not a definitive answer, by any means, but I'd reckon it has to do simply w/ gov't backing.
The .308 Win, aka 7.62x51mm NATO, was accepted as a more or less international cartridge. We had the M14 chambered in it, and overseas there was the FAL and the G3 as well. Most of our allies used it in their machine guns as well. Subsequently, it started seeing use in sniper rifles purportedly for logistics reasons: easier to stock fewer different kinds of ammo, and if push came to shove, it *would* fire ammo taken from an M60 belt. Kind of doubt that happened much, but that's the line of reasoning I always heard.
Also as a result of the 'international' aspect, most of the countries involved w/ Palma agreed to use a standard cartridge, and for some political reasons I don't entirely understand, that defaulted to the .308/7.62 w/ a 155gr bullet.
So now we have the .308/7.62 being used for military sniping by the U.S. and other countries, U.S. HighPower Service Rifle out to 600yds, and the Palma teams beyond that. Also somewhere in that mix, I believe the early BR shooters discovered that the .308 was inherently extremely accurate at shorter ranges as well. Add all this up, and add to it that the .308 fits in a short action, and you have a pretty powerful momentum to use the .308 Win.
From what I gather, the .30-06 does out perform the .308 Win, but not by a huge amount. When people wanted to make a definite step up, they went for a .300 Win Mag for a larger 'bump' in speed vs. the .30-06 if they were going to have to use a long action anyway.
Not saying these are the only reasons, or even the 'right' reasons, but they are the 'logic' I've always heard touted.
YMMV,
Monte