What was most effective in convincing the American public that the war was not winnable?

We could possibly have won it but we didn't want to have to many of our soldiers dead. We didn't wanna sacrifice to many. -Lovey

Given the understanding prevalent in the US High Command and Political circles of what the struggle was in Vietnam, YES, it was unwinable.

Vietnam was seen as a Communist invasion of sovereign South Vietnam. This was a fiction - in reality, what was going on was a Civil War. In addition, the US viewed it as a Conventional forces conflict, when it was primarily a conflict of infiltration and guerrilla tactics.

Given that "winning" the Vietnam War would have required a complete change of strategy and political realignment, not to mention completely overthrowing the existing South Vietnamese government and replacing it with something much more fair (and popular with the people), I can't see how it would have realistically been possible to win.