I have dealt with the same confusion for a good number of years, however is it true that the contemporary taino identity is base on a revival movement of people reclaiming their indigenous culture, however it is not necessarily base on an unbroken link from the times before columbus. After the spanish arrival and the colonization of the islands, the spanish imposed their ways on the indigenous people and used, manipulated, reenforced, extended, and interrupted their ancient trade networks for their own purposes, eventually enslaving indigenous people from the mainlands of mexico, central and south america, and later on with enslaved africans. They were all working in the mines, encomiendas, plantations simultaneously creating a new universe of cultures. The generation of the free and enslaved were lumped as "indios" by the spanish and used "good christians" vs "rebellious cannibals" or "caribs" as a way to justify their enslavement. However, despite the devestation of indigenous societies, the indigenous people had to adjust and persist in order to survive. We have to remember the indigenous nature of resettlement, refuge, and resistance where indigenous people and their descendants connected their natural and symbolic environment to a new universe created and eventual controlled by the europeans. There are material and spiritual aspects that survive from the indigenous people of the caribbean, such as, housing, fishing, farming (conucos), cooking utensils, healing through plants and herbs used for curative purposes among others. The reason for the neglect of indigenous caribbean is a direct result of the colonial quest for new resources and conquest towards the mainlands, and the massive loss of indigenous life through, war, disease, slavery and force labor. However the substitution, displacement and/or disappearance of indigenous people and their culture based on traditional historical narratives is countered by their presence in the caribbean today. They are part of a multi-ethnic, multicultural caribbean dynamic of european, indigenous and african, but where the indigenous people are the protagonist. To answer your question, the indigenous people of the caribbean definitely survived and are not separate from the precolonial world through the multi-ethnic and multicultural caribbean we see today, and is very complex, not found in historical narratives based on early colonial documents. As for the Taino identity in contemporary times is nothing more than part of a revival and reclamation movement and has nothing to do with what actually transpired with the indigenous people of the caribbean. There is a great book I have that was published last year "The Global Spanish Empire: five hundred years of place making and pluralism", with capture two discussing the indigenous cultural persistence of the caribbean and you would be correct in saying that there was dramatic changes over a long timeframe. Anyway, I hope this answers your question and best of luck!