BTW, asaivam does not equal vaishnavam. Siva and Vishnu are not opposite. This term asaivam/Saivam isn't about Siva vs Vishnu.
Vaishnavism considers Vishnu to be Brahman, just as Siva is Brahman in Saivism, Shakti is Brahman in Shakti-ism. Brahman is an infinite quantity, far beyond the capacity of humans to describe. So it was deemed convenient to take one quality of Brahman and describe it as a deity, like Siva or Vishnu. After all, every point in an infinite space could be considered its center.