I posted a couple times at EF-JFKA how JFK authorized the anti-Diem coup, despite being told it would likely result in the Diems' assassinations, and the complete lack of Kennedy Administration efforts to extract the Diems. Indeed, CIA inquiries about extracting the Diems were ignored---Lodge appeared to not want a government-in-exile situation, preferring the Diems dead.
There is a recorded conversation between Ambassador Lodge and JFK, in which Lodge lays this out, and many other warnings.
JFK responds, in part, by calling Madame Nhu a "bitch" and a lesbian (she appears to have been heterosexual).
The tapes and documentation indicate that while the Kennedy administration did not explicitly order an execution, Washington officials and JFK were well aware that a military coup could result in the assassination of Diệm and his brother.
There were different standards at play in 1963, and JFK tried to have Castro assassinated several times, and authorized the Diems' assassinations. That is part of the historical record.
People forget that JFK was deeply instrumental to the founding of South Vietnam, and he committed 16,000 military personnel to SV to help it survive against communist terrorism---and JFK himself called it terrorism. Cardinal Spellman was an influential figure as well, and close to the Kennedy family.
JFK may have wanted to modestly reduce US presence in SV in late 1963, but he was not going to abandon SV.
To be fair, the record shows that JFK did not want Diem killed, and that he was stunned and dismayed when he learned of Diem's murder.
JFK's authorization of the coup was one of the worst mistakes of his presidency and proved to be disastrous for the war effort, which had been going well from mid-1962 until the day the coup was launched (as we know from released North Vietnamese sources). The North Vietnamese could not believe that the Americans had been so foolish as to allow Diem to be overthrown. The Communists rightly viewed Diem's removal as a huge gift to the Communist cause in South Vietnam.
As bad as JFK's decision to authorize the anti-Diem coup was, his decision to approve Averell Harriman's ridiculous, fatally flawed "neutrality" deal for a coalition government in Laos in 1962 was arguably even worse. Eisenhower correctly warned JFK that Laos was the most crucial battleground in Southeast Asia. Eisenhower even told JFK he should deploy U.S. troops to secure Laos, since Laos was the key gateway for Communist supplies going into South Vietnam.
The North Vietnamese ignored Harriman's deal from the get-go and never withdrew their troops from Laos, just as Ambassador Nolting and General Harkins warned Harriman would happen. Thanks to Harriman's pathetic deal, the North Vietnamese were able to expand the Ho Chi Minh Trail through Laos, enabling them to increase the flow of supplies to the Viet Cong in South Vietnam (and later to North Vietnamese forces in South Vietnam). In fact, U.S. military personnel in South Vietnam began referring to the Ho Chi Minh Trail as the Averell Harriman Memorial Highway.