@KeshavSrinivasan has already given a fantastic answer on the origin of the practice, but I would like to address the purity aspect which you brought up.
Everyone can in fact receive the Sadari, but there are differences in the way it is presented. Most people who arrive at the temple in everyday clothes will receive the Sadari placed once on their head (note that there are not any mantras recited by anyone during this process).
However, it is different for those who arrive in Swarupam or Panandathiruman like this:
The placement of 12 thiruman on the body may be done after taking part in the ritual of Samasrayanam which includes the branding of the Sanga and Chakram, or conch-shell and discus, on the shoulders. Samasrayanam represents initiation into Sri Vaishnavism. This ritual can be performed by an acharya on anyone, regardless of caste.
If you arrive at a temple in this manner, the Sadari is placed first on the head, then on the right and left shoulders at the spot of the brandings, and then once again on the head. In some temples they might shorthand it by simply placing the Sadari twice on the head, or they might extend it by additionally placing it on the praying hands before it returns to the head.
You can think of it as an extra reward for those who have undertaken this extra step of commitment towards Perumal.