During WWII, one of the problems that the various allies faced was incompatible ammo, from pistols all the way up to howitzers. The U.S. sent quite a bit of equipment via lend-lease to various allies, many of whom were challenged to make good use of the equipment because they were not a standard calibre.
For example, the U.K. used 7.7mm rounds, where as the .30-06 is 7.62mm in diameter. We could send the U.K. .30-06 machineguns, but then they had to keep straight which units were equipped with them vs. their own 7.7mm ones. The major problem is in the supply chain. Same problem when we sent them 105mm howitzers. Their standard howitzer was the 25 pounder (87.6mm). Accidentally sending 105mm ammo to units that needed 25 pounder (and vice versa) caused several problems.
NATO decided from the very beginning to standardize on the ammo that they would use. .308 was chosen because the bullets themselves were practically the same as .30-06, but they were smaller (the gunpowder was smaller, but more powerful, making it equivalent in terms of kinetic energy) and lighter weight. Lighter weight is very important for troops in the field. If two bullets are otherwise equal and one is lighter weight, that's a no-brainer. The M14 was essentially an updated M1 Garand (with a more conventional clip rather than the stripper clip) in 7.62mm NATO. All 7.62mm sized guns, including snipers rifles, were standardized on the 7.62mm rounds. One can argue that the .30-06 or .270 may have been more powerful, but only slightly so, and standardization was deemed more important. The ability to have interchangable ammo with your allies should not be under valued.
America actually drug it's feet on pistol ammo. We stuck with .45 long after we had, begrudgingly, agreed to 9mm Parabellum. It wasn't until the 1980's that we finally switched to 9mm. America actually switched to 5.56x45mm ammo in spite of the fact that it was non-standard as far as NATO was concerned. NATO later adopted it as standard. 7.62mm NATO is still standard for snipers rifles, as well as light machineguns.
NATO standards were voted on by NATO, so even if we were pulling for .30-06 (which we were not, we were pushing for .308), we may well have been voted down. We were voted down on .45 ammo for pistols. BTW, .45 has become the standard for silenced weapons, largely because it is subsonic, unlike 9mm, which makes it ideal as you don't have to slow the rounds down (breaking the sound barrier does make noise).