Avvaiyar is an honorific associated with several Tamil poets. That matters whenever a popular story, a moral verse and an ancient poem are placed together. Their shared name does not automatically establish that all belong to one person or one period.
The story presented here is the Murugan tradition associated with Pazhamudircholai, also recounted by Tamil Nadu Tourism. It is introduced as a temple legend. We can follow its meaning while keeping the question of a precise historical identity open.