Thiruvaaraayirappadi is a work created by Bhagavad Ramanuja through Pillan. The lineage of Pillan belongs to Vadakalai sect. In the tradition of kalakshepam Pillan’s disciples were Engalazhwan, Nadadur Ammal etc. Thenkalai sect has lineages of Embar, Bhattar, Mudaliandan etc.
There was only one script written by Pillan on Thiruvaaraayirappadi. Bhagavad Ramanujar gave back that script to Pillan and appointed him as one of the head pontiffs. As Desikan comes from the lineage of Pillan, he gained it. As there was only one copy, and it was not available to the Thenkalai sect, they had to create 9000 padi.
Swami Desikan has not specifically referenced 9000 or 24000 padi. His work Nigamaparimalam is not available for us. In few of his expositions particularly on Thirumaalai, he has quoted Periyavaachaan Pillai’s writings.