Firepower and portability, in a nutshell.
.223 is a damn effective round out to about 200 yards and the vast majority of combat engagements happen within this distance and usually much closer. Since a soldier can carry roughly twice as many rounds of .223 as an equivalent weight of .308 this is an important feature, as the same individual can lay down double the rounds or last twice as long before needing resupply.
Full-auto controllability was also a factor since shooting any of the .308 main battle rifles (M14, FAL) on auto was next to uncontrollable, just wasting ammo. Shot to shot follow up is also much faster and accurate with the smaller caliber as well.
Obviously the .308 is a better round in many respects: penetration of cover, longer effective range, etc... But using that argument you could say why not a larger caliber than .308, like the .338 Lapua? I'd make the .308 look positively anemic, punching great big holes in things 1000+ yards away, is less affected by wind, blah blah blah.
But the .338 would be an awful battle round because it's too heavy and powerful for the average guy to tote around and use effectively.
The whole thing is a balancing act since there is no perfect cartridge fore every situation and the military feels the .223 is the best all around. A lot of us here feel the same way. (but we loves dem .308's too!
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