They became unpopular when we started obviously losing,
Wrong once again. The military was NEVER losing. The NVA and the VC got their asses handed to them routinely. Don't bring up TET because you'll only be highlighting your ignorance. Militarily, TET was a catastrophic loss for the NVA and VC that the US media managed to turn into a brilliant victory for the North.
The US lost when the politicians failed to allow the military to prosecute the war as a war. (If that's confusing, think Gulf War 1). As the politicians continued to fuck up the process, the populace lost faith, lost interest. Then the US withdrew with the expected result.