What evidence suggests that the Tainos were not annihilated and to some extent still exist today?

Some of the Tainos married with Spanish and Africans so they still exist. The few survivors intermarried with other ethnic groups.

Some of the Tainos married with Spanish and Africans so they still exist to some extent today. Most of the Tainos died because of disease but there were a few who survived. The genes of those who survived are now probably mixed with the genes of others since their numbers were too little to create a Taino population again.