What does the lender do with this package? The numbers are analyzed and verified. The borrower's ability to pay the mortgage is calculated with these new numbers, usually showing the hopelessness of the situation. The lender then derives a value for the property and compares that value to the amount due on the mortgage and the offer from the purchaser.
With the average cost to a lender for foreclosure running around $50,000, there is some incentive for them to consider a short sale offer if the numbers work. The may use a AVM (Automated Valuation Model) or a BPO (Broker Price Opinion) or both to determine the price at which they could sell the property in foreclosure.
If the lender sees the possibility of recouping more of their investment from the short sale route, they may very well approve it. After all, the process will be less costly, and it will be completed sooner than a foreclosure. The lender's assets to hold reserves to back up this loan will be freed up sooner.
We're back to your original package here. Many times lenders will commission a real estate broker to do a "drive-by" BPO. They never even enter the home. And computer AVMs do not show the current condition either. So, if there are repairs that will be necessary to get the home sale-ready for foreclosure, then you want the lender to know about them. Be complete with repair quotes in your package.