the primary reasons 5.56 was chosen over 7.62 was due to uncontrollable recoil in full auto, along with the weight of the rifle and ammo.
I would say that the uncontrollable recoil was the primary
excuse used, but not the real primary reason.
Gen Curtis LeMay wanted the AR15 for his SAC guards. He was ok with them using the M1 Carbine, but the M1 Carbine was going away. The Army was getting rid of it, and at that time the Air Force got its small arms and support for them (spare parts, tools gauges, etc.,) from the Army.
The General was in a bit of a pickle. The Army's infantry rifle (M1 Garand, and then the M14) were bigger, heavier, and more powerful than needed for Air Force SP's /SAC guards. The M1 Carbine, which was enough for the role, was going away. He was introduced to Stoner and his ARs, and found the AR15 (which was at the time a .222Rem) He thought it was just the thing to fill his needs, and showed it to a number of folks in the govt (including JFK) to gather support and get the brass to allow him to buy some for his troops.
The "whiz kids" of the MacNamara Defense Dept. fell in love with the rifle and pushed it to be the new standard infantry rifle. Some push back from some people in the Army who didn't want it caused the creation of the 5.56mm (.223 Rem) round, and with that round and the AR rifle the M16 was born and it was ordered to be the new infantry (and general use) rifle.
(yes, I've left out quite a bit, there are entire books on the subject)
The reasons most often given were the M16's light weight, controllability in full auto fire, and double ammunition for the same weight compared to the 7.62NATO. Additionally, the military was desperate to obtain a counter to the AK-47 that wasn't an AK-47.
The 18lb BAR took us through WW II and Korea, and wasn't thought to be uncontrollable. The 9lb M14 firing essentially the same bullet at the same speed was another matter, and the high cyclic rate of the M14 compounded the problem. The BAR (usually) ran at around 550rpm. The M14 is 750rpm.
SO, with half the weight and nearly half again a faster firing rate, is it any wonder it was "uncontrollable"?? What amazes me is that there HAD to have been people in the development process who could recognize this, but apparently none of them could get the brass to do so.
I can no longer recall the source, or I would share it, but I heard that either in the late 60s or 70s some civilians tinkered with the M14, and were able to reduce the cyclic rate down to around 5-550rpm and found it much more controllable in full auto fire.
The Army never got the chance to do that. Literally it was orders from on high saying warehouse the M14s and use the M16 now. And the M16 has its share of problems as well, if people remember. Some were deliberate, most weren't. We've spent decades fixing those bugs, and finding new ones as military demand for modifications to better suit certain missions created them.
Now they are looking at some as yet only vaguely defined 6.8mm caliber and their wish list for what they want it to do is ,,,simply amazing to me, and some other folks.
Perhaps they think that since they want to get to the moon, putting out requirements that would get them to Mars (if they could be met) will result in people trying, and while they will fail, what they will accomplish will be enough to get them to the moon. I don't know but it seems that way to me.