Why would you pay off the mortgage on a short sale?

The idea behind a short sale is that the house is not worth the mortgage balance, so that the bank allows the owner to sell the house for less than the mortgage balance (hence the term "short")  

There is no magic in buying a short sale 

And indeed a short sale is not necessarily great - usually a short sale home is priced at market value. A short sale is really only benefits the seller (unless a sale of real distress) and discomfort are many (turn around the long times, etc.).