Most of the people in that list were likely not Tamilian - Shankara was likely Malayali, Kumarila & Dharmakirti are just as likely to have been from Assam, and Madhva was Tuluva.
It literally quotes the Azhvars... you're right about Malva, that was a mistake. I don't agree with a possible assamese origin of either Kumarila or Dharmakirti, and see no reason to doubt Taranatha, and in the 9th century, the distinction between Malayali and Tamil is academic.