It does seem that Nixon might have been an entirely different person and President had he been elected in 1960: No 1962 CA governor's race loss and meltdown, no eight years worth of building up his paranoia about the Kennedy's.
If elected in 1960, Nixon may have felt driven to "measure up" to Eisenhower, especially since he felt slighted by Eisenhower and his people. That desire might have made for a much more statesman-like Nixon. Also, at the time Nixon was a much more centerist Republican (despite the "Pink Lady" campaign) and, considering some of the things he did even in the post-Goldwater time of his actual Presidency (like China, creation of the EPA, consideration of a national health-care plan, etc.), he may have been a much more progressive President than one might initially think.
The only point I might debate is re Vietnam. It's nice to think that someone (anyone) may have seen the folly of sending in the troops, but Nixon in the early 1960's may have been so cognizant of the "who lost China" arguments of the previous decade that he may have felt forced to fight it out to "save" Vietnam from "going red".
One other thought: Rather than picking Johnson to run for Veep again in 1964, I wonder if Humphrey (or whoever) might have picked a Southerner who may have (in the context of Johnson not having become President and pushing for the CRA and VRA) seemed more progressive. Maybe an Albert Gore Sr. or William Fulbright?
Carl Sanders on a Kennedy ticket in 1968 is an inspired selection.