After WWII the US military wanted to modernize their weapons after seeing the German weapons. Congress wasn't interested in spending money on weapons right after winning the world's largest war and the military was downsizing and getting rid of extra weapons. Winchester has lost most of the military powder contracts to Dupont (IMR). Winchester still had friends and contacts in the military and the Cold War was just starting.
During WWII there were shortages of copper to make brass and lead to make bullets. Winchester in a bid to make military contracts showed the military that it could about match military 30-06 in 300 Savage using their ball powder(within 100 fps). The US military used this fact and the Cold War to tell Congress that a new "smaller" cartridge would require less copper and lead. In fact the first 7.62 NATO bullets were steel core without any lead. The military wanted the full 30-06 velocity and less shoulder angle than the 300 Savage. So the 300 Savage cartridge was extended and made in to the 308 Winchester.
Of course this "new" cartridge would require "new" weapons to use it. This is one of the reason the military was so interested in replacing the 30-06. It gave them a good excuse for new weapons that they wanted since WWII. The Winchester ball powders at the time produced much more muzzle flash than Dupont's IMR powders. This resulted in the long flash hiders on the M14, M60, FAL/L1A1 and MAG58. It wasn't until later that it was discovered that slotted flash hiders didn't have to be so long and didn't require a small opening at the front. The long slotted flash hiders did work better than the older cone flash hiders.
Winchester thought that since the new cartridge used their ball powder and they designed it that they would have exclusive contracts for it. Dupont still had military and political friends and contacts. The increase in size in the 308 Win from the 300 Savage allowed some Dupont IMR powders to work within specs in the 308 Win. Dupont was able to have the military ammunition contracts for 7.62x51 to be split between Winchester and Dupont (IMR/Remington).
During the Vietnam War the ball bullet was changed from steel core to lead core. There hasn't been a lead or copper shortage since WWII. Winchester was interested in contracts and the military was interested in new weapons. The 7.62x51 is more of a 300 Savage sized up than a 30-06 sized down. The base cartridge for 300 Savage was 30-06 though.