The M 16 problems are kind of a tall tale in some respects
Units like the 1st Cav who trained and deployed with the M16s had very few problems because they had used them prior to deploying in 1965. They also were a very strac unit and had good weapons discipline. Units like the 1st Div were given their M16s in country and had zero training about how to use them and no equipment to clean them. A team of vet NCOs sent by Colt to assess the issues found that the weapon maintenance for the division was horrible and that all weapons were poorly maintained. That is a unit problem not an equipment problem.
American units who didn't have M16s used the M14. I was there in 1966-67 and M16s were in short supply with almost all going to mobile Infantry battalions, the rest of us just used our good M14s which were wonderful weapons. SF and advisers may have carried Carbines and Navy personnel (like Jim) used them by choice. But it was never an issue of only the M16 or carbine. By 1969 or so everyone had M16s.
It is my understanding that the Marines resisted giving up their M14s with a vengeance and considered the M16 to be a poor substitute.