A brief history of how the M 16 came to be the US service arm:

It was the USAF that first adopted the M16 for use by their security troops as a replacement for the M1 carbine. In the mid 60's the USAF bought something like 50,000 M16s for this purpose. Some of these were the first ones used in Vietnam about 18 months ahead of any issue to the US Army.
The Army decided to adopt the M16 based on the success of the USAF with the weapon. But, the Army ordinance department insisted on a couple of changes in design to meet their peculiar needs. One of these was that the powder used be switched to the standard Army "ball" powder. This propellant had a slight advantage in power for the number of grains used but left considerably more residue in the barrel and other components than the powder the USAF used which was basically equivalent to civilian granular powders.
It was this change that caused most of the problems with the M16 in early service. Since the USAF had been using the rifle for over 2 years in quantity with few problems the Army put the M16 in service with the ordinance changes but with little testing. The result was all the jamming and other problems with the gun in early Army service in Vietnam.
As far as the round and gun go, the US Army predicated their choice on weight and rounds carried. A soldier could carry only so much weight. If the rifle he carried was lighter and the round carried was lighter he could carry more rounds or other equipment as a result. On paper the 5.56 round with its tendency to tumble on impact was supposed to have equal stopping power to the 7.62 NATO round. So, on paper a soldier with an M16 could carry more ammunition of equal effectiveness to that of the M14. The kill an opponet versus wound them is just a spicious an argument. At the tactical level wounded are evacuated as possible not in preference to terminating the firefight. That is if the enemy is firing on you and someone on your side gets wounded it is not going to immediately require others in your unit to stop fighting and help the wounded man. The reality is more complex as I previously noted.