I don't think cutting off right-wing "Democrats" in the South was a bad thing.
This was accomplished irl when the Kennedy Administration embraced civil rights legislation in 1963.
Kennedy, on the other hand, very much campaigned in a right-wing manner. His main difference from Nixon was that he would escalate the Cold War from Eisenhower's levels.
Kennedy ran to Nixon's "right" on foreign policy in the campaign, but this didn't mean anything. e.g., Kennedy attacked Eisenhower for not deposing Castro, but the Republican Administration was plotting his demise behind closed doors. There would have been no difference on Vietnam, either, except that Nixon may have escalated the war faster- the seeds of this policy were planted under Eisenhower (and Truman), after all.