A Cuban historian named Sergio Guerra has an interesting and, to my mind, convincing thesis on this. In his book 'El Dilema de la Independencia' (The Dilemma of Independence), he basically puts Hispanic American countries in one of two categories. First, the classical cycle of countries were loyalist Junta governments became the incubation stage of full blown separatist revolutions. Second, countries where local elites sided with colonial authorities to suppress Juntas when and if they arose and only backing independence when a) separatist armies brought it by force or b) they saw laws being passed by Spanish Liberals as a threat to their authority and privileges.

Cuba would fall into the latter category.

When Fernando VII and the rest of the Spanish Bourbons were overthrown in 1808, the machinery of the Old Regime government crawled to a halt or, as was the case of the Consejo de Castilla (Council of Castille), were coopted by advancing Napoleonic troops. This lead to local authorities being overthrown and replaced by Juntas (councils) which then organized resistance against Napoleon. (Miguel Artola: La España de Fernando VII)

In the Americas, these struggles for power between paralyzed traditional authorities (whose last orders were basically 'don't do anything to antagonize the French') and those supporting proactive Juntas had the added dimension of being colonies.

Mind you, Spanish colonies weren't colonial in some aspects. The legal and institutional framework which held them together was more feudal than modern; they were more 'far flung possessions of the Spanish crown' than they were colonial possessions of the Spanish State and its people. In practice, there was discrimination against creoles and there were limitations on their trade, but Spain was not to enact full on modern colonial policies which legally discriminated against creoles until the 1837 Liberal Constitution, which comes after our period. A contemporary figure who emphasized that Spain's colonies weren't technically colonies was the priest Fray Servando Teresa de Mier.

In the case of Cuba, who have the extra dimension of the country having a massive number of slaves. After the Haitian Revolution, Cuba's relatively limited number of slave plantations grew exponentially. This demographic shift combined with the tales of white survivors of the Haitian Revolution led to fear and paranoia over a slave rebellion.

Around 1808-1809, Havana developed a Junta, just like most of Hispanic America did. But while the Juntas in Caracas, modern Venezuela, and Buenos Aires, in modern Argentina, survived, the one in Havana was swiftly crushed by colonial authorities. I haven't studied the crushing of the Junta in a great deal of detail, but what is clear is that local elites were reticent to counter this action with a full blown rebellion.

As the years dragged on, cases like Venezuela showed that Spain was willing to basically destroy everything, including white planter elites, and go so far as to unite with non-white slaves and free people of color (see Jose Tomas Boves) as long as they could suppress separatist rebellion. Between the threat of Haiti-style slave rebellion and the apparent willingness of Spain to side with the slaves against their white masters, colonial elites in Cuba were paralyzed. (Olga Portuondo: Cuba. Constitucion y Liberalismo. 1808-1841)

This fear was reinforced when the Aponte Rebellion took place in 1812. An apparent alliance between free people of color and slaves, inspired by the Haitian Revolution, was discovered and brutally suppressed. Slave rebellion was a regular pattern in Cuba for centuries, but this was dangerously well articulated and apparently went beyond a bunch of resourceless slaves killing one master on one plantation. (Matt Childs: The 1812 Aponte Rebellion in Cuba and the Struggle against Atlantic Slavery)

This underlines the 'dilemma' of independence I mentioned earlier. Sergio Guerra states that the dilemma that colonial elites faced was how to emulate an American style political revolution without falling prey to Haitian style socio-economic revolution.

Now, elites certainly played both sides and some seemed to support independence. During both Liberal periods in Spain during the independence wars (1812-1814, 1820-1823), conservative leaning colonial elites in Cuba, Mexico, and Peru became attracted to separatist thought. Cuban elites sent out feelers to the US to see if Washington was willing to annex the island and thus keep it safe from foreign invasion and domestic rebellion. Peruvian elites played with the idea but in the end only became independent when Simon Bolivar invaded and liberated the country from Spain (it should be noted that Peru's elites had brutally suppressed their own separatist factions earlier, so it wasn't an issue of there being no separatist inclined Peruvians). Mexico ended up becoming independent through a conservative military coup which aimed to make the country independent from Spain, but keep as much of the status quo as possible, going so far as to ask Spain to sent a member of the Bourbon dynasty to sit on their throne and (failing that) declaring themselves the Empire of Mexico under coup leader Agustin de Iturbide.

In Cuba's case, their inability to resolve the 'dilemma' of independence without risking being destroyed in the process led them to keep their options open, but largely side with Spain. Cuba became a key source of income for Spanish attempts to reconquer the rest of the Americas (sugar and coffee plantations were booming at the time) and the key staging grounds for forces meant to reinforce loyalists on the mainland.

There were certainly plots to make Cuba independent, such as the 'Soles y Rayos de Bolivar' and 'Gran Legion de la Aguila Negra' conspiracies in the 1820s, but these were discovered and suppressed under Captain General Francisco Dionisio Vives.

So Cuba's case is closer to Peru's in that the only way that independence would come was if an invasion occurred. The US helped stop that. Both Colombia (ie. Bolivar's Gran Colombia, not the smaller modern Colombia) and Mexico tried to prepare and send invasion fleets against Cuba, but domestic troubles, economic collapse/debts, and open opposition in the US (John Quincy Adams made it clear we'd military support Spain) led to the failure of those expeditions being launched.

Despite being one of the key proposals of the Panama Congress in 1826, domestic issues and American opposition sank the project before the ships were even launched. It should be added that Great Britain also, though more quietly, opposed the proposal and let Mexico and Colombia know it.

By the 1830s, mainland Hispanic America was so embroiled in its own domestic issues (economic depression and collapse, massive debts, constant rebellions/coups, etc.) that there wasn't enough will power or resources to even attempt another invasion. During the same period, Spain became embroiled in economic collapse and the beginning of the Carlist Wars which would devastate the country for decades.

Now, regarding your last question, the question of what you mean by independence is important. The first mention of Cuba being, on its own, and independent Republic seems to have first been suggested by Felix Varela in the 1820s but appears to have had few adherents. Most Cubans who wanted independence from Spain were pushing for joining another country, whether the US, Colombia, or Mexico. The Cuban flag is based off of the one used by Narciso Lopez in his 1851 invasion of the island; the reason it looks like the American flag is that Lopez based it off of the Texas flag with the objective that Cuba, like Texas, would eventually join the USA. Cuban elites talked up annexation in private, secret associations like the Club de La Habana (Havana Club). Even the Ten Years War (1868-1878) began as a project to make Cuba a part of the United States. Only after they were once again rebuffed by the US, which was understandably reticent to annex a territory which still had slavery and thereby reopen recent wounds, did they support insular independence over joining another country. And even then, separatists didn't necessarily represent a majority of the population until much later.