Though the word Craig used was wrong his point was correct. The US government supported Batista and his thugs due to American business interests there.
After Batista was kicked out and Castro came to power they actually looked at the US for recognition. When the Congress rejected his overtures to them he turned to the Soviets, starting down the path of what would eventually lead to the embargo and missile crisis.
I re-read my post in light of your comment and will clarify.
When I wrote it, I was thinking strictly in terms of the Cuban history related to the trade block, i.e., the Castro revolution.
When I said what their independence is related to, I meant in contrast to virtually all other Latin American countries in the period who have relations relatively co-operative with the US as the dominant nation, to varying degrees, I mean independance from the US influence, 'doing their own thing' as far as independent from the US unlike others, and attempting with varying degrees of success to encourage other nations to be more independent of the US as well.
(Some might ask, but wasn't Cuba not independent, but a Soviet satellite state? I'd say no - they had an alliance of convenience and Cuba was expected to show some level of political support for the USSR in exchange for the support it got, but it was always primarily an independent state and even criticized the USSR at the height of the cold war, albeit with constraints. They used each other.)
But, I can see how the term 'independence' could be confusing, since it's most often linked to the end of the Spanish role. On the other hand, I don't want to change the word from 'independence' to 'revoution' because I think 'independence' better captures the topic of independence from US influence after the revolution, while revolution is in regards to the specific Batista regime that happened to have US backing.
I agree with you that Castro wanted a US relationship; I'm not familiar with a lot of the details but my impression is that it was a combination of error in the US approach - this was in the 'any ally, however much a brutal right-wing dictator (Batista was not the worst at all), is a good partner while we oppose anyone at all 'left' period, under Eisenhower and the Dulles brothers (who infuriated Churchill with their refusal to accept any peace with the USSR); and that Castro wanted a leftist approach the US just was not ready for.
Castro was not that reasonable a person to reach peace with. At a time we were backing brutal regimes who partnered with US corporations to exploit the countries, we weren't even close to allying with him with his leftist economics that would not be too friendly to exploitation by US firms.
This mistake not to reach some peaceful arrangement would have seemed a lot worse if it had actually led to the nuclear exchange we came closer to than most realize in 1962.
For a note on the relaitonship with the USSR, Kruschev infuriated Castro by agreeing to withdraw the missiles without any consultation with Castro, who heard it about it from the news IIRC. On the other hand, Kruschev thought Castro was nuts based on Castro's saying to him privately that the independence from the US was important enough an issue that the nuking of Cuba was an acceptable price to pay and nuclear war was worth it to not let the US have dominance (including with their missiles on the Soviet border).
I wonder what Castro would say later about that position, I haven't seen any comments. It's good to remember though that Castro wasn't alone, many in the US similarly had too low a bar for justifying nuclear war, including some at the top of the Pentagon, who Kennedy thought were about as nuts as Kruschev did Castro.