Why did JFK pick Lodge? Short answer: Probably because, in part, he was a Republican and could give JFK political cover in case of failure.
From Reeves' book: Kennedy was inclined to send Edmund Gullion, who was finishing a difficult tour as Ambassador to the Congo, as Nolting’s replacement. He was the young consul in Saigon who, twelve years before, had told Congressman Kennedy that the United States would be driven out of Vietnam if U.S. troops moved in after the French colonial army was defeated. But Rusk was opposed to Gullion and Kennedy decided to let him have his way. Besides, he already had another idea. At a ceremony at the Pentagon, he had run into Henry Cabot Lodge, Jr., the man he had defeated for the Senate in 1952. Lodge, the Republican nominee for Vice President in 1960, was a major general in the Army Reserve and had been doing his annual month of active duty. They had talked for a couple of minutes about Vietnam, and Kennedy remembered that when the first American combat troops had been sent there in 1961, Lodge had said he would be willing to serve as ambassador.
A couple of days later [i.e., June 15/16], the President asked Secretary of State Rusk to ask Lodge, unofficially, if he was still interested and available for the Saigon posting. Lodge said he was and Kennedy invited him to the White House. “Cabot,” he said, “I am beginning to spend more of my time on Vietnam than anything else. The Diem government seems to be in a terminal phase. . . . I’d like to persuade you to go out there as Ambassador and as my personal representative.” He would be reporting to the President rather than to Rusk. Once again, as with Lucius Clay in Berlin, Kennedy wanted a Republican in place to share the responsibility for whatever was going to happen next. That, the President told Lodge, might well be the overthrow of Diem and his family—probably in a coup by his own generals. With the Americans looking on, approvingly.
Appointing Lodge is one thing; deferring to him so much is another. Note, he would be reporting to JFK rather than Rusk! How did Rusk accept that?