Amount you can carry
Fighting between the big-round and small-round groups reached a peak in the early 1960s, when test after test showed the .223 Remington (M193 5.56×45mm) cartridge fired from the AR-15 allowed an eight-soldier unit to outgun an 11-soldier unit armed with M14s at ranges closer than 300 meters. U.S. troops were able to carry more than twice as much 5.56×45mm ammunition as 7.62×51mm NATO for the same weight, which allowed them an advantage against a typical NVA unit armed with Type 56-1s.
https://en.wikipedia.org/wiki/7.62%C3%9751mm_NATO#Adoption_in_battle_rifles
Interesting chart there too about how much ammo you can carry in a 10kg load of magazines.
260 rounds of 762 NATO
620 rounds of 556
360 rounds of 762 Soviet.
Long Range Effective Accuracy
On the other hand, once you get out to like 300 yards/meters, lighter rounds will get tossed around by the wind, but personally I was having success with very heavy 85gr Open-Tip Match 5.56. But that's premium stuff, and standard 55gr and 62gr projectiles were very hit or miss shooting prone at a 8-inch target.
Recoil and Quickness of Follow-up Shots
.308 out of an AR-10 feels like twice the recoil of 5.56 out of an AR-15. It's not fun. The gun weight too feels like twice as much. If I'm lugging something around I know which I'd prefer.
What are your thoughts? Also, is the new larger US military rifle/caliber contract won by SIG Sauer dead in the water? I think it is, except maybe for specialized roles like DMRs.
If you think .308 is too much recoil and it makes it unpleasant to shoot, it's 100% a skill issue. Not to say you suck if you have a problem with it, but literally it is only a problem if you are holding the rifle wrong. Even slightly wrong and it's a problem. So train more, basically. It's easy to get bad habits when you only shoot 5.56.
The extra weight of ammo and the rifle itself is the problem, not the recoil. At any reasonable combat range, .308 has little advantage over 5.56. In combat, you absolutely need at least one person with something in .308, either a dmr or a machine gun (ideally both), but for the majority of your people 5.56 is far superior.