I'm old enough to remember the Vietnam War and the AFL-CIO of the day was staunchly anti-communist and pro-war. Most of the opposition to the war before Nixon became president came from the pacifist left and student led organizations like the SDS.
I already cited the AFL-CIO’s conservatism. Under McCarthyism, most real leftists had been purged from the leadership of large unions. What you say about the students is true, but an incomplete picture. If you want a better one, check out Philip Foner‘s US Labor and the Vietnam War.