I wrote graduate research on small arms development and procurement in the early cold war in the USA (1945-68). The small caliber high velocity concept gained an awful lot of traction when the AR15 was tested in the early 60's in Vietnam. At the time it was being considered as a platform to arm the small stature ARVN forces that were still using ww2 and Korean War surplus like M1 and M2 carbines. The data collected by ARPA (Advanced Research Project Agency) showed the ammunition had massive wound capabilities. They were getting reports from advisors (remember that the USA was not the primary combatant yet) that the round was killing about 90% of those hit.
So when the M14 production was halted before procurement was anywhere near complete due to massive issues with contractors like H&R and we found ourselves in the mix officially, we had no other platform to really turn to but the M16 and the 5.56 round. Of course we made many small changes including altering the twist from 1 in 14 inches to 1 in 12 inches to pass muster in cold weather accuracy. But between the ARPA reports, the weight savings, the increased control over the M14 in full auto, and the belief that it was ultra reliable it is not hard to see why someone like Robert MacNamara, the ultimate bean counter, was happy to support the switch. He was not alone given Curtis LeMay had pushed for the same gun and ammo a few years earlier.
If you can find a copy, Ed Ezell's "The Black Rifle" does a good job of showing how the switch came about and the debate. Plus it has cool pics of prototypes and hard to find documents.