You'd really have to nail down what the criteria is you're judging "best military [small arms] weapon" by.
The criteria The History Channel program went by as I recall had things like: numbers of rifles manufactured, length in service, reliability, accuracy, etc.
I think the AK-47 and a couple others scored so high because of the reliability, numbers mfg'ed and years of service.
I think that the M1 Garand had to be right near the top though with it the one small arm weapon that arguably had a big part of winning WWII, plus M1A and M-14 are "direct descendents" of the Garand.