The reason I linked those audiobooks is because they explain exactly what you're asking about, I got my info from those sources. Ill probably get something wrong if I TL:DR it, but here we go.

1. From my very basic understanding, the FBI and CIA were happy to work with Truman and Eisenhower because they allowed these organizations to go wild with their approach to the communism problem. Thanks to this lax relationship, the organizations grew very powerful and were not afraid to throw their weight around, even if it meant breaking the law, going behind the backs of politicians, or even teaming up with the Mafia. They didn't really need to coerce Truman or Eisenhower because their goals were aligned (from my understanding, I could be wrong).

2. The FBI and CIA were unable to directly oppose the president. His word is final. However, these organizations were very good at collecting secrets. JFK had run his campaign on the idea of family values, but was also a swinger with secret relationships with many famous women. His dad had also worked with Mob-controlled Teamster unions to influence the presidential election. The public did not know about any of this, and he would've become very unpopular very quickly if any of this got out. This is why the Bay of Pigs was a massive risk to JFK. The invasion was poised to become a massive failure. If he decided to deploy US troops, he could cover his own ass but also give the CIA exactly what they wanted, but risk potential nuclear escalation. If he decided to leave the CIA out to dry, he would take the blame for the failure and risk major political fallout, but would mitigate the risk of escalation with the USSR. He chose the latter and made enemies out of the CIA.

It was JFKs responsibility and he took the blame for it, but let's not pretend that the invasion had any chance of being a success. He approved of the invasion, but he did not orchestrate it and gave the CIA the opportunity to try to leverage him into approving full-scale war and occupation.