Perhaps it might be an idea if you actually watched the documentary as I was merely reporting what the documentary makers reported. And given the praise the series has garnered, I don't doubt most of its accuracy. [Note - you talk of my lack of understanding of US history. Why get personal? They were not my ideas. They were those of the film makers!]
First point to note is that I only bracketed Kennedy, Johnson and Nixon because it was under their Presidencies that the war was conducted. Indeed it was under Kennedy that it effectively started as far as the US was concerned. But I agree with you – I think! I do not believe you can bracket Kennedy with the other two. Yet you cannot help wondering how Kennedy, having got in and given his words at the time which I come to later, would have got out. It is an fascinating problem.
The fact is, though, that the Vietnam War started long before the US got involved - in any capacity whether by government action or secret CIA involvement. And the start had absolutely nothing to do with communism - absolutely nothing! The desire of Vietnam to have independence was a movement against a vicious French colonial power. As soon as the Japanese surrendered, Ho Chi Minh proclaimed independence, using some of the phrases from the US Declaration of Independence. On several occasions he had earlier tried to get Roosevelt and Truman to help get rid of the hated French colonial power. After all, the USA was staunchly anti-colonialism. Sadly for what was to happen, both turned Ho down and allowed the French back into Indo-China. The French did not like the Vietnamese and most of their soldiers did not want to be there. I cannot recall how many commanders they had in a small number of years, but the last one who planned the stand at Dien Bien Phu and promised the French who had by then grown weary of the war it would result in total French victory, committed suicide when he realised his plan had been a total and utter disaster.
Kennedy toured Vietnam in 1951 prior to the end of that war with the French. At that time he certainly did not believe that military force could overcome the forces commanded by Ho. Russia was soon to start feeding ordinance to the North and China was to follow. So from that point onwards Vietnam did indeed became a Cold War pawn. But only from that point.
You appended an article to your post. Interestingly, it stresses the following –
After the signing of the Geneva agreement [in 1962], the leaders in Beijing seemed more willing than their comrades in Hanoi to accept the fact that Vietnam would be indefinitely divided.
As for Kennedy, none of us has any clue how he would have treated the war in Vietnam had he lived. You'll recall his inaugural address,
"...we shall pay any price, bear any burden, meet any hardship, support any friend, oppose any foe, to insure the survival and the success of liberty."
South Vietnam fell into the group of friends. Eisenhower had warned Kennedy he would have to send in troops. Kennedy wrote to the South Vietnamese President later that year, "the United States is determined to help Vietnam preserve its independence..." - this despite South Vietnam being to all intents and purposes a murdering dictatorship. He then started to send military advisers to the South. The first 400 were soon increased by Kennedy to over 16,000, as you point out. He also sent in troops to pilot the helicopters he sent over, thus involving Americans in combat operations. He justified this by saying it was
"to prevent a Communist takeover of Vietnam which is in accordance with a policy our government has followed since 1954."
He also stated in a television interview with Chet Huntley in September 1963,
"What I am concerned about is that Americans will get impatient and say, because they don't like events in Southeast Asia or they don't like the Government in Saigon, that we should withdraw. That only makes it easy for the Communists. I think we should stay."
"I think we should stay" is surely pretty clear!
Kennedy was pressed by his advisers to get out of Vietnam, the more so when Buddhist monks started to immolate themselves in Saigon. To suggest that this was caused by communist actions is utterly laughable and totally untrue. It was a demonstration against the corrupt, murderous dictatorship of President Ngo Dinh Diem and his loathed brother Ngo Dinh Nhu. In this protest the Buddhists were joined by the South's important Catholic community,
Given Kennedy's stated views, it is hard to believe he would have got out of Vietnam and let the North take control. Initially Congress would probably not have permitted it. Those in power in the USA were still in thrall to the China lobby and the constant refrain of "Who lost China to communism?"
Perhaps he thought his approval to the coup which led to the murder of the Ngo bothers less than 3 weeks before he himself was assassinated would lead to a change in the south that would lead it to finally take greater control of its own defence. In fact it threw the south into even greater turmoil. But what he might have done will always remain one of life's great mysteries. No one can state anything as fact.