The Pedersen Device used a .308 caliber bullet in a rimless case much like the .32 S&W long. You are probably thinking of the .276 Pederson, a cartridge developed by the same J. D. Pederson who invented the Pedersen Device. It was, however, a genuine 7mm cartridge, using .284" diameter bullets. The .276 reference was to the bore diameter.

At the time Winchester first offered the .270 there were two other cartridges then in existance, the .256 Newton, a 6.5mm cartridge based on a slightly shortened .30-'06 case, and the 7X64 Brenneke, a cartridge almost identical to the .280 Remington.

Possibly because of accidents caused by firing WW I 8X57 ammunition in Winchester Model 95's chambered for the .30-'06 cartridge, Winchester seems to be overly concerned with the possibility of firing the wrong ammunition in one of its rifles. The .256 Newton could be fired in the .270 with no harmful effects, except to accuracy. The .270 could be fired in a 7X64 rifle, again with no harm, except to accuracy. Winchester may have simply wanted to avoid the "millimeter" designation, which seems to be the kiss of death to American cartridge nomenclature (with the glaring exception of the 7mm Remington Magnum.)

This is probably the explanation for Winchester's decision to market its own design of .30 caliber magnum cartridge, the .300 Winchester Magnum, instead of using the obvious choice of the .30-.338, a cartridge which came into being shortly after the introduction of the Winchester Magnum trio of .264, .338 and .458, all based on the .375 H&H Magnum case shortened to 2.5" to function in a standard length action. Winchester's thinking was possibly that they didn't want their ammunition in use in combination with a wildcat chamber (one which was similar enough to the obsolete .30 Newton that it was sometimes referred to as the .30 Belted Newton). Whatever the reason, the success of the .300 Winchester Magnum soon eclipsed that of the .30-.338 and it's sister, the .308 Norma Magnum.

Unlike Remington, which actually chambered 40X target rifles in .30-.338, Winchester has never embraced wildcat cartridges. Rather than adopt the original version of the .220 Swift, known as the Wotkins Original Swift, made by necking down the .250 Savage case to .22, Winchester chose to neck down the 6mm Lee Navy case and create an entirely new cartridge, which would not fit any existing chamber.

Remington has never hesitated to embrace cartridges Not Invented Here, as its lineup of .22-.250, .257 Roberts, .25-'06 Remington, .260 Remington, 7mm-08 and .35 Whelen all attest. It's just a different philosophy.