(and during the late 50s and early 60s a very easy mistake to make, perhaps impossible not to make given the position of the US in the world after WWII: : the only intact and strong Western nation left with the Soviets the other and they looming and having taken half of Europe + other places already.)
The other part that we were in a "PIT" is also not factual: during Eisenhower and JFK it wasn't a pit. It was 4 years later. In '63, it was money and US training with a relative handful of operational troops. That's where the line should have been drawn. Aid to the extent it was given until '64-'65 was about right if you understand the world at that time and didn't have the hindsight of 50 years down the road or even 4 years down that road with the massive US troop build-up in '67 under Johnson.
So, keep in mind JFK was dead and not responsible for the Johnson and Nixon era vis a vis their policies for Vietnam. And some signs are he wouldn't have let it go there - again, no one knows.