- The US backed a coup that put Fulgencio Batista in power
- Batista murdered over 20,000 Cubans (not my words, that's what John F. Kennedy said)
- Cubans were in horrific poverty exactly because of US policies
- When Cubans overthrew the dictator who was murdering them by the tens of thousands, the US attempted to invade them, the most likely aim being another US-backed dictatorship
The US embargo is purely sour grapes over losing access to a country's natural resources it wanted to exploit coupled with, as you point out, a dysfunction in the way US elections work. Pointing nukes at a ruthless, amoral enemy with vast resources was a perfectly valid reaction for Cuba given the context.