I do think that it might have prevented the pro-war backlash (which, contra the OP, was really more a creation of the '70s than the '60s - certainly there were groups like Students for a Democratic Planet on the interventionist left and the Minutemen on the right in the 1960s, but they didn't really get off the ground until the McGovern presidency), and thus things like the riots at the 1976 DNC (fun fact - I know one of the delegates there, who still swears up and down that McGovern actually won, but the DNC didn't count votes for him on the last ballot because they were worried that if Scoop Jackson lost, his supporters would burn down St. Louis, which, yeah, probably) and the interventions in Greece, Angola, Nicaragua, the Philippines, and so many other places under Agnew and Finch (and Kirkpatrick and Dornan).
Really, as popular as Johnson is now, that's mostly a product of post-World War III historiography. When your country caused, through a toxic cocktail of malice and incompetence, thirty million deaths directly and probably twice as many indirectly through famine, disruption, and the actions of "emergency governments", and has itself lost a few major cities and seen a generation plunged into shellshock and guilt, it's natural to hold up the one president who really did beat swords into plowshares, and who managed to convince the American people to go along with it (unlike McGovern, who for all his good reputation really was in the wrong place at the wrong time, and made a number of unforced errors). But the thing is, not much of his original legacy survived - Medicare and Medicaid were gutted by Finch and not re-established until the Richards era, Agnew ate civil rights law for breakfast (despite Dornan's well-deserved legacy, he did actually make a lot of progress there), Kirkpatrick (herself a fellow Democrat!) turned the peaceful Space Race into the military space weapons program, and so on. Politically, it was easier for postwar leaders to ascribe their projects to Johnson rather than say "hey, here's a brand-new idea I came up with" - even when, as with the University of the United States or the Family Aid Acts, they really didn't have much to do with LBJ.