a war that nobody but a small number of politicians seemed to want
This is revisionist nonsense. Vietnam was the most popular war in US history. At the time of the Tonkin Gulf Resolution, 90% of American's supported deeper involvement. No other war has ever had so much support. For instance, only 70% of Americans thought the 2003 invasion of Iraq was a good idea. Of course, support for any war declines as it drags on, especially if we appear to be losing. But it is a lot easier to get into a war than out, so it is only the support at the beginning that matters.
The Gulf of Tonkin Resolution passed the Senate with 98 votes. The two senators that opposed it were both voted out of office at the next election. It is silly to say that this war was forced on the American people by the politicians, when the truth is that it was fear of the voters that pushed the politicians into supporting the war.