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.