So I think you might just be looking at the wrong dates. Vietnam was from roughly 8 March 1965, with the first official deployment of about 3500 marines in a small defensive deployment, increasing to around 200,000 troops in country by the years end. The war ended in roughly 1972 (officially 73, but most troops were pulled out during 72), with the closing days of vietnamization seeing the slow withdraw of US forces and funding. Some of the most iconic fighting, and therefor most likely a good portion of what you have seen in films and documentaries, happened during the Tet offensive in 68, or afterwords, due to the much larger dry season offenses in the second half of the war.
Now, here is where I think you are being mislead by dates. The M16a1 was fielded initially in March of 1965, and replaced the M14 as the Standard-A issued weapon for ground combat units in 1967, with the M14 being designated a limited issue weapon. By the end of 1967 all army and marine combat units in Vietnam had been issued the new M16a1 rifle, previously designated the XM16E1 rifle. In fact, by December of 1968 the USA had shipped an additional 600,000 M16s to equip the allied Vietnamese forces as well.
The date you are using is the probably based on the date the army officially removed the limited standard issue supply of M14s from circulation and use in Basic Training and European stations, and replaced those with the M16 as well. The Army also replaced the weapon as the standard Service Rifle in late 1969, so that could also be the date that is mixing you up. The army has many standards, but the only one that really matters is fielding dates, and those are 1965-67.
So the M16 was almost exclusively used as the standard rifle in Vietnam by combat troops from 1966-67 and after. So it was in fact the standard weapon for a majority of the war, and was the standard weapon during a majority of the actual fighting.
Hope that helps clear things up for you and answered your question!
Some sources for further reading are below.
Army review panel for the M16 Rifle in Vietnam.
TIME Magazine article on the M16 in 1967.
The Black Rifle: M16 Retrospective, Collector Grade Pubs. 1987 ISBN-10: 0889351155.