No we wouldn't have, it would have been 8 years of Nixon, followed by California Governor, Ronald Reagan in 1968.
Vietnam, the 1960s shift, the 1965 Immigration Act, the domination by Ted Kennedy and the Kennedy family, none of that would have happened.
If Nixon had taken office as the proper victor in 1961, it would’ve altered the dynamics for the GOP for the decade. Since Goldwater would not have challenged Nixon as an incumbent in 1964, there would’ve been no massive GOP losses that year (conversely, there also would’ve been no Southern breakthrough at that time, either). 1966, as a 6-year election, would’ve probably been bad for the GOP (similar, but hopefully not as bad as 1958 was), but Reagan might’ve lost challenging Pat Brown that year (and had he lost, would’ve never been a Presidential contender).
Indeed, the 1968 standard-bearer for the GOP would’ve been Vice-President Henry Cabot Lodge, Jr. Had JFK still been in the Senate, he would’ve been the nominee in 1968 and probably would’ve beaten Lodge (as he had in 1952) as Conservatives would’ve been very unhappy (even having Goldwater as the VP running mate to try to assuage them).
The big question by then would’ve been if Nixon had gone all out to have gotten an early and decisive win in Vietnam and reclaiming Cuba (and executing Castro and Guevera for crimes against humanity). It certainly would’ve been a helluva dynamic alteration going into the ‘70s.