They fled from a country that’s been communist for over 60 years and they’re not big fans of communism you say? Weird.
A lot of the established Cuban community in Miami are the children of wealthy Cubans who fled during the revolution. So yeah they fled communism but maybe not for the reason you meant.
The country hadn’t been communist for 60 years when they fled though, they weren’t fleeing poverty. They were mostly pro-Batista affluent people and corrupt government officials who went into exile in Florida expecting to return when Castro was overthrown. Castro was never overthrown, so they stayed.