If JFK were in today's Democratic Party, they would run him out just like they are with Lieberman.
You got to remember that the parties then are different than the parties now. In reality there were really 3 parties back then, the northern democrats, the southern democrats, and the repubs. You could easily argue that there were 4, i.e. the liberal white shoe northeeastern repubs i.e. Nelson Rockefeller, John Lindsay etc. and the other repubs i.e. Barry Goldwater. Richard Russell of Georgia and Hubert Humphrey were members of the same party in name only :)
Anyway when JFK was probably blackmailed at the LA convention to put LBJ on the ticket and throw Symington overboard, it was if someone from another party was on the ticket. The Kennedy clan *hated* Johnson and marginalized him to the point that in 61, 62 it became a running joke. They would send him to BF Egypt for some assistant deputy minister's wife's cousin's funeral just to keep him away from the corridors of power. Also most folks think that in November of '63 the announcemnet that LBJ was going to be removed from the ticket was coming soon. Technically they were in the same party but by today's standards they weren't.