To expand on this. Nixon made it a priority, though it took some time because of politics, to fix the Draft, which was done by 1971, which removed many of the policies that had made it so unpopular earlier (mainly the deferments). Yes, while some of the anti-war movement was tied to the Draft, it was more related to politics.It was an anti draft movement as much as, or even more than, an anti war movement. That the United States now use professional soldiers is why the general populace doesn't much give a damn how many wars are fought or for how long.
Universities aren't exactly centrists in the US, since the 50-60s there was a major shift as more European educated professors of the Frankfurt School came to the US to teach, who used the institutions for indoctrination in the long march through the institutions, where they established havens of far left ideology, which of course spread to student bodies, as that was the point. The loudest protests originating from the student bodies (aided by various professors and administrators) came from groups like Students for a Democratic Society (SDS), who were very far left leaning, their own charter positioned them as "anti-anti-Communism". They organized a lot of the opposition, which spread to a lot of the rest of the youth, tying it with the Counter Culture movement that had been brewing since the early 60s, and the Civil Rights movement. So it was simply popular to be anti-war, at least if one was young. As my father said, "To be against the war meant you were hip." The media was on board too, especially younger generation. It created a loud minority, but overall most Americans were still pro-war, these were later referred to as the Silent Majority (who got Nixon elected).
The big political shift occurred leading up the 1968 Democratic Primaries, where numerous contenders who had previously been pro-war, read the tea leaves of popularity, noticed the shifting winds of the Democratic Party, and that became a major party platform for the party who'd started the war. This worked out well for them, especially after Nixon won the presidential election, as they were his opposition, they controlled both House and Senate in Congress, so could impede Nixon as much as they wanted, which is just how politics work (all sides do that). Impeding Nixon who was trying to fix Vietnam means sabotaging him (similarly to how Republicans sometimes tried to do the same to FDR during WW2, its just politics).
Nixon won in '68 running on a Law and Order platform, appealing to the older generation who were quite sickened by the chaos happening in America due to the counter culture and civil rights riots and mass protests, where the summer of 1968 saw most US cities suffer major rioting and arson. Nixon also wanted to end the war in Vietnam, but done victoriously, by finally allowing the US military to take the gloves off, not completely, but removing many of the insane stipulations that the LBJ administration had been doing (like telling North Vietnam exactly where were going to bomb, so they could move civilians out, where as they moved out anything of importance and moved in SAMs). Nixon also took the war to the safe havens, Laos and Cambodia, which should have been done early in the war but LBJ didn't have the guts. He also lost the criminally stupid McNamara, and reshuffled MACV leadership, which saw major changes for the betterment of the conflict, as the generals running it under the LBJ administration, specifically Westmoreland, were chiefly responsible for us losing the war through their incompetence.
Worth watching to showcase what role leadership plays in warfare: Westmoreland: The General Who Lost Vietnam
Nixon's administration, far from perfect, was still on much better straights than LBJ. The National Liberation Front (Viet Cong) had broken themselves in the Jan '68 Tet Offensive. North Vietnam was weakening and finally having their supply line and logistics in North Vietnam targeted. Vietnamization and other COIN policies replaced the asinine Search and Destroy/Body Count policy used under the LBJ administration. The South Vietnamese, far from perfect, at least were holding together in the calamity of a mixed major communist insurgency plus a foreign invasion originating along their entire border.
But by then, too many Americans, specifically those who weren't fans of Nixon, were against the war. They wanted us out, they didn't care if South Vietnam fell to communist, to them that wasn't a big deal as many had some sympathy already for various left wing ideologies. So we left. And then Saigon fell.