Council of Elrond

According to the rules, discarding used (tapped) site cards after leaving, they are put in your discard pile, right?
After exhausting your play deck, you have to take the site cards out and return them to the location deck.
Wouldn't it be easier to use a 'second' discard pile only consisting site cards that were discarded?
Thanks for answering!

It would be more convenient, right.
However there is reason to keep discarded non-haven site cards in discard pile.
It decreases a chance of finding a removable card by Aware of Their Ways.

Thanks for your quick answer.