Your Majesty: Presumably JFK's medical conditions don't get exposed? Remember, before Kraus and Burkley saved him from a wheelchair IOTL, he was using Travell as the primary-care physician. She was resistant to the point where she was only retained to prevent her leaking her files to the New York Times. Even RFK IOTL, nearly nine years younger and incomparably fitter, had three or four stress-induced illnesses. How is Jack going to cope if he needs crutches and descends Marine One and AF1 one step at a time?
It depends on how much you think his condition would deteriorate and in what ways his condition and doctor situation would be amended (some twists of fate to get him in contact with doctor's and physicians who warned of Travell in similar form and led to similar results is possible).
Depending the situation and if it is allowed to go that far (crutches or a wheelchair), it'd be handled very carefully. The same sort of press technique of the Kennedy clan football games to show off an athletic President in apparently good and even suburb health would be used to show him as fully capable and mobile, probably in a similar fashion as was done with President Roosevelt in hiding his own condition. If someone does a timeline, this is a rather interesting possibility.
Politically: How is JFK going to distinguish himself from the Nixon administration, both being relatively centrist?
Likely similar to how he had in 1960, save with the element of the outsider to boot. I don't think I'd refer to either as a centrist, but rather highlight the limited distinctions between the parties in this era. Before a Liberal party and Conservative party, there were big tents. As the Democrats represented the Blue collar and Republicans the white collar, Kennedy would probably play to the common man, minorities, unions and so forth. The issue here, however, is that he would run with talking points derived from what had been done the past 8 years, which is something we can't be sure of and which we are obviously discussing currently. Similarly, it depends on the GOP candidate for 1968. If it's Reagan for example, Kennedy can likely swamp him in the number of minority voters he'd get as Reagan's Conservatism wasn't a friend of Civil Rights.
He'd be experienced, but they could use the OTL anti-Nixon argument: he lost it for us in '60, someone else should try.
That would obviously be an issue. But, he'd have built himself as a grand statesman, a prime representative of Democratic party, and so forth over the last eight years as Nixon had done on the opposite side. He'd also probably avoid the embarrassments that plagued Nixon in that time. Had Nixon not failed to win California in 1962 (a move which almost destroyed his career and which made it appear he wasn't going to return in 1968), I would posit Nixon would have had an easier chance in the nomination. And of course, he could use the message as Nixon did of "I'm the best you've got".
Well, two political scientists did an analysis that showed both parties were judged equally CR-friendly in 1958-9. MLK Sr. even initially endorsed Nixon. Then came the two Kennedy phone calls and that finished that idea.
On Civil Rights, depending on how the GOP goes and if the Conservatives do rise, the GOP could lose the remaining black vote they had as they did in 1964 and over the course of the years following. If I recall, there remained a bit of division among the black populace. If I recall correctly, FDR had brought a good deal of blacks into the Democratic party due to his welfare programs and so forth, and many said something to the effect of "Lincoln really is dead". However, many remained with the Republicans, who did not have the Dixiecrats, and who retained the support for Civil Rights. Then came Goldwater and the rise of the Conservatives, and then went the black vote. That was the point where MLK abandoned the Republican party (keep in mind, as a Southern black, King and his ancestors had seen a GOP which supported his people, and the Southern Democrats who were none too friendly to the black population, and supported the GOP for the Progressives and against the Conservatives. I address this because too many modern Republicans cite King's being a Republican as if support for Conservatism or the party as it is now, when nothing could be further from the truth). If 1968 sees a Conservative run the GOP ticket and/or the Conservatives rise, a similar thing is probable to happen.
Re "50 states": that was stupid. If he'd focused on the swing states like OH, PA, IL, IN Nixon would have won. Even the debates needn't have been a problem if he'd used the sunlamp like he did in anticipation of facing Bobby in 1968 IOTL.
Wasn't Nixon recovering from some sickness during the debates, though?
Re deviousness: Nixon was just the candidate most comfortable with Rovian tactical maneuvering for its own sake.
Nixon took deviousness to an extreme, and a rather scary one at that. If people are planning on assassinating a reporter who speaks out against you under your administration (as did happen under Nixon's OTL administration; in fact, there were a number of other controversies or whatever you want to call them under the Nixon administration which were just as bad as Watergate, and sometimes worse) it shows signs that you've gone to a high point.
It is fun to consider what other Watergate like acts a Nixon administration of the 1960's could do.
But on the deviousness, LBJ and RFK were also big fans of tapping, dirty tricks and RFK's purposeful neglect of civil liberties in regard to MLK's phone cost him the Oregon primary when Hoover leaked it IOTL. Though none of them ever planned assassinating a journalist, which sounds positively Marcosian. Now a TL where PI becomes a US state and Marcos becomes POTUS- Wilson and PATRIOT would look like peanuts.