Winchester actually designed a .28-caliber round based on the .30-06 case before the 7x64 Brenneke appeared (which most sources agree was in 1917), and a bolt-action rifle to handle it. In the years just before World War One, they intended to introduce both the rifle and cartridge in Europe. Called the .28 Winchester Centerfire, it used a .287" bullet. But the war killed the project.
As to exactly why Winchester chose .277" as the bullet diameter for the .270, nobody really knows, but one educated guess is they didn't want the Mauser 7x57's that had poured across the Mexican border during the Pancho Villa years to be rechambered for a higher-pressure round, so made sure the bullet was just small enough to prevent that.
The reason 7mm (and some other calibers, such as 6.5mm and .30) ended up with fast enough twists to stabilize the heavy-for-caliber spitzers so fashionable today is they originated as military calibers when smokeless powder first appeared--with very heavy round-nosed bullets. The twists had to be fast to stabilize those bullets.
But purely sporting calibers such as .22, .25 and .27 ended up with slower twists. The theory back then was not to over-stabilize high-velocity bullets, because any flaw in the bullet balance resulted in poor accuracy--and those three calibers came into use after lighter, higher-velocity spitzers became the trend for hunting.
“Montana seems to me to be what a small boy would think Texas is like from hearing Texans.” John Steinbeck
