The events that preceded the coup against Ngo Dinh Diem were essential because they convinced the policymakers in the United States that Diem had to go. It was because of these events that President Kennedy, for example, was willing to allow the coup to happen.
There are at least two general events that made the US willing to allow the coup. First, the war was not going particularly well. It was clear from what was happening that the South Vietnamese army was not capable of fighting the communists. It was also clear that it was not very well motivated. Second, and more importantly, there was Diem’s horrible handling of religion in South Vietnam. Diem’s repressive actions against Buddhism in a majority Buddhist country were foolhardy in addition to being morally suspect. The fact that Diem was willing to do these things, and the fact that his brother and sister-in-law (closely linked to his regime) would say the things they did, clearly showed that Diem’s government would never be able to get popular support in the South.
Because the war was going badly and the Diem regime looked to be beyond saving, Kennedy approved of the coup. The events preceding the coup were essential because they convinced Kennedy of the necessity of approving the coup.