That IMO is where things can really change: if Kennedy loses the Vice-Presidency in 1956, he isn't elected President in 1960 (if ever).
I'll agree wholeheartedly with the notion that if Kennedy is the losing VP candidate in 1956, he won't be elected president in 1960 (indeed, he may not even be a candidate). I'd suggest, though, that he'd do something analogous to what FDR did IOTL: build his credentials on the sidelines as senator and bide his time for a future run--perhaps 1968 or 1972, assuming his health would permit that. Those dates might move up depending upon the prognosis of his Addison's disease.
Sidebar I: without Kennedy in the mix for '60, that race among the Dems will be fascinating. You'd probably have Lyndon Johnson slugging it out with Hubert Humphrey for openers--with, say, Adlai Stevenson on the sidelines as a possible compromise candidate. None of them, of course, is friendly to the prevailing southern school of thought about race relations and civil rights, so southerners may feel shut out to the point of possibly bolting (Smathers for president, maybe?). The Dems are going to have a knock-down, drag-out battle for the nomination that would likely yield some fairly hard feelings and divisions (the losers may be lukewarm about support the winners, perhaps) such that the GOP nominee--presumably VP Nixon--would have a much less difficult time than he did IOTL. So...probably looking at a Nixon presidency from 1961 to 1969, which in turn means a significantly different Nixon than we knew.
Sidebar II: a Nixon presidency in these years likely obviates the southern strategy, and the Goldwater wing of the GOP remains relatively small. At the same time, Rockefeller's recent divorce might well shelve him as a candidate also. Wouldn't be surprised if Bill Scranton would be the 1968 GOP candidate. If so, then you'd have two relatively young (both approximately 51) attractive, articulate moderate candidates, assuming Kennedy gets the Dem nomination.
That might take honors as one of the most issue-based, thoughtful elections in decades. And the butterflies from that are the size of B-52s.