Easy peasy.
JFK's health deteriorates in 1959, to the point that he is confined to a wheelchair/crutches (as he was at times when undergoing--or about to undergo--medical procedures, like when he was given the last rites in a London hotel in the late forties.) He doesn't fully recover before the primary season, and the newsmedia isn't willing to give him a free pass as per OTL with his Addington's. This is the first election covered by handheld sound-on-film cameras, you just can't do an FDR at this time.
And without the testing board of the primary elections JFK isn't going to risk going to the convention as a delegate-less candidate.
In this scenario you have to justify Humphrey or Symington being able to beat LBJ at the '60 convention. I don't see either having the moxy to pull off a first ballot victory, and that radically increases the chances of LBJ's strategy getting him the nom.
Johnson has at least as good a chance at beating Nixon in the general as Kennedy did. Perhaps even better.
Yes, he was a Southerner, he was an outsider, he was considered anti-liberal and ultra-sectional by many liberal Democrats.
But he was also an incredible talent. Hell, he was more telegenic than Nixon, something we forget because of JFK's intense charisma. And money isn't going to be a problem for his campaign--that's the factor that he might use to get the bosses to support him over HHH & Symington at the convention.
IMO his biggest challenge will be getting liberal support, and stomping out any talk of a third party candidate to his Left.
Johnson/Humphrey? Johnson/Harriman?