So I am not sure of the setting for Europa Universalis 4, but I assume the setting is probably the early modern period, 1400-1800-ish?
If so, then yes the Caribbean was economically productive and dynamic at that time. Over the course of the 17th and 18th centuries the Caribbean became a global center for the production of two important commodities— sugar and coffee. These two goods were part of a larger consumer revolution in terms of diet, especially the consumption of exotic colonial goods. The Caribbean was an ideal environment to produce both of these goods and therefore attracted a great deal of overseas investment as well as ancillary industries such as shipping, milling, construction, and of course this was tied into the broader trans-Atlantic slave trade with all of its broader entanglements. During the 19th century the production of sugarcane and coffee shifted to the South American mainland while competitors popped up across the colonial world in South/Southeast Asia, the Pacific, and West Africa, as a result the Caribbean lost a lot of its economic vitality and importance.
During the early modern period, cane sugar shifted from its status as primarily an expensive elite commodity to a good that was widely available for consumers of nearly any rank. This was due in large part because of sugar cultivation in the Atlantic and then in the Caribbean resulted in lower prices which led to increased consumption. As anyone with a sweet tooth or food science background will attest, sugar is addictive. Once people had a taste for it there was a sustained market demand for affordable sweeteners. The primarily sources for colonial sugar in the Caribbean were Saint-Domingue (Haiti, French Empire), Jamaica (British Empire), and Cuba (Spanish Empire), with an entire constellation of other islands and colonies that contributed sugar as well as agricultural goods that supplied and sustained the slave laborers on the larger sugar islands.
Coffee also transitioned from its status as an exotic good that could only be acquired through trade in the Eastern Mediterranean to a colonial product from the Caribbean. If we take the case of France as a good example— coffee may have first appeared in 1669 when an envoy from the Ottoman Empire introduced this new steeped beverage to the royal court and the the Sicilian Franceso Procopio Cutò may have opened the first coffee shop as early as 1686. It took a while for the beverage to catch on— it was exotic and considered potentially unhealthy as it might unbalance one’s humours with its bitter taste and tendency to cause heart palpitations. However, by 1700 coffee was caught up in a wave of demand for exotic luxury goods from the East— coffee became fashionable and its consumption was gradually localized to include the addition of milk and later on sugar. Still, early coffee houses were typically adorned to reflect French stereotypes about the Ottoman or Safavid Empires while coffee was consumed out of porcelain cup ware, another exotic import.
To acquire coffee, the Compagnie des Indes received a crown monopoly to import the beans from Ottoman ports to France as part of its normal trade with the empire. This potentially lucrative market for coffee quickly attracted competitors despite the royal monopoly. Colonial officials on Réunion in the Indian Ocean began planting arabica coffee and grafting it with local wild varieties of coffee and saplings were sent to France’s Caribbean colonies to attempt planting as well. There was a long drawn out legal battle between the Compagnie des Indes over whether coffee could legally be imported from the colonies (as well as whether it still counted as coffee or should be considered a different pharmacological substance), but in the end consumer demand, which encouraged smuggling and rule bending, won out and coffee began to be imported form the French colonies. The first cargos of coffee from Saint-Domingue arrived in 1738 and by 1790 Saint-Domingue supplied 60% of all the coffee sold on the global market, nearly all of it going to France. In 1790, on the eve of the Haitian Revolution, coffee exports were valued at about 71.5 million livres. This was still dwarfed by the value of sugar from the colony ~200 million livres (raw & refined sugar combined).
An important aspect that might be missing from the game though is the economic cost of sustaining this industry as well as the intense competition the developed from South American colonies, such as Brazil, which came to dominate sugar production and eventually dominate global coffee production in the 19th century.
In terms of cost, one of the side effects of the sugar and coffee industries in the Caribbean was the introduction of malaria and yellow fever as well as the Anopheles and Aedes mosquitos that spread each of those diseases respectively. These diseases had a massive economic cost especially in terms of human lives. The annual death toll from disease, which was made even worse by the sort of infrastructure needed for sugar plantations (the creation of ponds and swampy lands which were ideal breeding sites for the mosquitos), was a huge drain on the profits reaped in the Caribbean. Though, as Vincent Brown argues, death also drove the economy of the Atlantic as the death of enslaved laborers required the shipment of new slaves, soldiers, sailors, etc., the creation of new burial grounds, the shipment of bodies and personal effects back to Europe, or commemorations and memorials.
tl;dr - While the Caribbean had a great deal of economic significance, especially for the production of colonial goods, "death took a divided" to quote Brown.
Julia Landweber, “‘This Marvelous Bean’: Adopting Coffee into Old Regime French Culture and Diet,” French Historical Studies, Vol. 38, No. 2 (2015): 193-223.
William Gervase Clarence-Smith and Steven Topik eds., The Global Coffee Economy in Africa, Asia, and Latin America, 1500-1989, (Cambridge: Cambridge University Press, 2003).
Michel-Rolph Trouillot,“Motion in the System: Coffee, Color, and Slavery in Eighteenth- Century Saint-Domingue,” Review 3 (Winter 1982): 331–88.
Vincent Brown, The Reaper’s Garden; Death and Power in the World of Atlantic Slavery (Harvard: Harvard University Press, 2008).