This is a two part answer. First, re: Bay of Pigs, second, more generally, re: Cuba since.

The short version re: Bay of Pigs is that Cuba was an especially hard target, previous coups were always huge gambles even against weaker targets, and the US government made decisions that made the already huge gamble an even weaker bet.

Slightly longer version.

As opposed to many Central and South American targets, not only was the Cuban government extremely popular with strong legitimacy, but any opposition to it was marginalized and divided. Many opponents had already emigrated by 1961, other political parties had been coopted or dissolved by the time of the invasion, neighborhood watch committees that would send intelligence to the government (the CDRs) were monitoring potential opponents on a local level, and the military was firmly on the side of the government. One of the key lessons that people like Ernesto "Che" Guevara took from the 1954 coup against Jacobo Arbenz, for example, was that the military needed to be unquestionably loyal to the government. There could be no independence, no chance that it would rebel. The army was still small by 1961, but it had had enough time to grow from the motley crew of a couple thousand guerrillas in 1959 into the nucleus of a real army. Add to all of this that Cuba is an island, not a country with porous land borders, and this is a hard target indeed.

This brings us to point 2: even previous coups were huge gambles. If you look at previous coups, like the one against Jacobo Arbenz, in some histories it's easy to feel like the CIA is this inhumanely competent and skilled adversary. While we don't want to overcorrect in the other direction, Tim Weiner's excellent Legacy of Ashes leaves clear that even Operation Success against Arbenz almost didn't work out. The success of previous operations in Iran (1953) and Guatemala (1954) makes some feel like one against Cuba along the same lines in 1961 could only have failed due to the Kennedy administration's added mistakes (we'll get to that in a second). In practice, all of these were huge gambles, and only a few years before Cuba the CIA tried to pull off a covert operation in Indonesia in 1958 only for that project to fail, partly (irony of ironies) at the hands of a US trained military force. This was always a high risk, high reward, style operation that had a solid chance of going south in the worst of ways.

Finally, Operation Zapata (1961) was originally begun under President Dwight D. Eisenhower and then fell largely into Kennedy's lap as a fait accompli which he then tweaked. He could have gone back on it, but doing so would have made him seem weak on Communism, so in order to avoid the domestic political cost of calling it off, he tweaked it multiple times in his first months in office, including insisting that the US military would not be spearheading the invasion invasion itself, that it had to be by Cuban exiles. What happened and why, what the different players in this mess assumed would happen, etc., is still debated, but Piero Gleijeses makes a strong argument in his article Ships in the Night: The CIA, the White House and the Bay of Pigs for the read that in addition to assuming Kennedy would buckle, nobody really thought through what was supposed to happen after the initial invasion. The shift of invasion site from Northern Cuba near a major town to the middle of the Zapata Swamp, for example, made the invasion way easier, but the backup plan for the force to flee to the Escambray mountains (where there was still armed peasant resistance to the government) now no longer made sense because it would have involved a long trek East across choke points the Cuban government could easily take ahead of time. Another issue was that the initial air raid that was supposed to take out the whole Cuban air force, giving the invasion the air superiority it desperately needed, failed to get all of it, and failed to come back in time to clean up what had survived the initial bombing run. In short, a bunch of unforced errors.

There's a really good popular history podcast called Blowback which recently did a whole season explaining the details of the invasion, what went wrong, etc. You can find a link here: https://blowback.show

Some good bibliography:

Gleijeses, Piero. "Ships in the Night: The CIA, the White House and the Bay of Pigs." Journal of Latin American Studies 27, no. 1 (1995): 1-42. Accessed July 31, 2021. http://www.jstor.org/stable/158201.

Rasenberger, Jim: The Brilliant Disaster: JFK, Castro, and America's Doomed Invasion of Cuba's Bay of Pigs

Weiner, Tim: Legacy of Ashes: The History of the CIA

Now, why did the Cuban Revolution survive after that? That's an even more hotly debated question. The Cuban government's firm control of the military, all political and social organizations (except the Catholic Church), and strong popular support combined with a media monopoly helped keep it together in the 1960s and 1970s. It was a really bumpy road, and the coalition it forged in 1959-1961 has slowly frayed/collapsed or outright passed on since, but even today it still has something of a base, between true believers (increasingly small group), people who have done well for themselves in the government and have material reasons to continue to support it, and people who are critical of the gov but want reforms not an outright destruction of the post-1959 project.

One really good read that should help explain how the Cuban government lasted the Cold War is Lillian Guerra's Visions of Power in Cuba: Revolution, Redemption, and Resistance, 1959-1971, where she gets into things like how near total government control was, now everyday people actively participated in maintaining government control for various reasons, etc.

I wrote a long summary of how things stood by the late 2010s here: https://www.jacobinmag.com/2018/06/miguel-diaz-canel-bermudez-castro-cuba

I think one key factor, in addition to all the others already talked about, that's central to understanding why the Cuban Revolution survived so long when many others didn't is that it had a strong nationalist base of support. The Cuban Revolution is talked about in terms of Communism, which it also is, but I don't think that's actually the core of its message. It was a nationalist revolution that framed itself as a path to both national sovereignty and prosperity. While it couldn't provide the latter, it could still sell itself as the Latin American project that managed to keep itself free from American imperial control. Whether or not you agree with that idea or framework, that's something many of its supporters will still invoke to you if you talk to them privately. That this is a project about Cuba being able to be free from American control. The US embargo, attempts to undermine its government, etc., not only provide excuses for the government to explain its failure to succeed, but also help hold together the government's frayed but still important coalition. If the options are seen as a deeply flawed Cuban project vs a US imposed post-Revolution government, many will still choose the former. It's one of the great ironies of US foreign policy towards Cuba. The government hasn't survived despite US aggression. It has in no small part survived because of it.

I just finished listening to a podcast about the Dulles bros and this was thread was very interesting. Thank you so much 🙏🏼