...from Wikipedia:
"At the time of its introduction, the .276 Pedersen was a solution to a significant problem. The US Army wanted a general issue autoloading rifle that would fire the .30-06 cartridge, but such a rifle was prohibitively large with existing designs such as the Browning Automatic Rifle and French Chauchat. A weapon of the same weight as the M1903 needed to fire a smaller cartridge. Pedersen's cartridge was viewed as a compromise as it was underpowered compared to most military rifle cartridges. This decreased recoil energy made possible a reliable, lightweight semi-automatic rifle with existing technology. Despite these early problems with semi-automatic designs, Garand's design was eventually able to handle the .30-06 cartridge; the need for a lighter caliber dissolved. The Pedersen rifle was unsuitable for the .30-06 and it, too, was dropped."
The Pedersen design involved a recoil-operated toggle-top bolt, much like a Luger pistol. It operated on a delayed-blowback principle instead of a locked breech design, which is probably why the standard .30-06 cartridge was too powerful for it. It also required dry-wax lubrication of the cartridges to enable reliable extraction of the spent cases, which the Army testing board found unacceptable. So while the rifle design was found unacceptable, the board was impressed enough with the .276 cartridge to continue working with it.
The only “serious” competitor the Garand faced was Melvin Johnson’s recoil-operated rifle of 1941, which actually did see some limited service with the Marines and OSS.