Some factors I believe should be taken into account:
Vatican obviously had a stake in the success [or lack thereof] of the presidency of the only avowed RC USA president.
No matter what JFK might have done had he not been shot, .... he initially increased the American involvement in S.Vietnam. Summer of 63, Buddhist monks were protesting against the Roman Catholic dominated Diem regime by burning themselves to death in front of Western journalist.
September 11, 1963 New York Times published an article by David Halberstam revealing RC aspects of the war.
I'm certain the Vatican did not want the American public to see the war as a war in support of a Roman Catholic dominated regime. The Vatican did not want the war to be tied to a RC president. JFK became worth more as a dead martyr than as a living president.
Head of CIA John McCone was a RC Knight of Malta,.... and many high level officials in key positions [government and media] were also avowed subjects of the "holy father" in Rome.