It seems likely that the actual reason for Johnson entering the war was neither the brush in the Gulf of Tonkin, nor the shelling of the American hospital, Gordon Tullock suggests:
I think Johnson simply saw that it would
make it much easier for him to defeat Goldwater if he stole from Goldwater his military position. In any event the sending of troops many of whom were drafted, and who were under command of a rather inept general, Westmoreland, rapidly developed serious domestic difficulties in United States for the war in Vietnam, but insured Johnson’s victory in ’68.
Johnson may have turned hard right to outflank Goldwater in ’64, but after that he attempted to move to the other flank against the New Left. Having disposed of his conservative opposition, he devoted a lackluster effort to Vietnam, but at the same time he promoted his Great Society expansion of welfare and social services.
Unfortunately for him, while the Old Right was riding into the sunset anyway, the New Left was gathering strength feeding off the media and cultural revolutions, and they would not be as easy to outmaneuver.
Despite all the political and military command ineptitude, Johnson presided over both the defeat of the NVA in the 1965 Battle of Ia Drang where American forces repelled an invasion from the north, and he presided over military victory during the Tet Offensive/Battle of Saigon where the Viet Cong were decimated and neutralized for the remainder of the war.
Those turned out to only be tactical victories. The North Vietnamese won the moral and psychological battles which were much more important. They convinced New Left operatives in the West to oppose the war and by doing so took Johnson out of the race.
The New Left mobilized on college campuses and mass media, eventually infiltrating the political establishment. Johnson countered with the greatest expansion of government handouts in human history. It wasn’t enough and he was defeated by ’68.