If you know anything about Cuba, you know that they're in the middle of a slow transition from Castrism to a more approachable Chinese model, with private ownership being progressively reintroduced. Normalisation of relations with the US is high on the agenda; the death of Castro will have to be managed carefully if established powers want to avoid trouble, and relaxed access to Miami is something they'd really like to offer the population in order to keep the peace. I wouldn't be surprised if a general or two had already agreed with US diplomats a protocol to follow right after the big man finally kicks the bucket.
At this point in time, it is quite credible that Cuban leadership wouldn't risk compromising their political outlook for the sake of a random American troublemaker. It would go a long way to explain why Putin took such a long time to let him in the country, after authorizing his passage (and even then, he likely granted asylum to Snowden only as retaliation for the hardening of US positions on Syria).