There's a process to getting the mortgage payoff statement. First, you'll need to contact your lender and let them know you want the information. Depending on your lender, you may have to sign in to an online account, call a helpline, or send a formal letter to start the request process.
Your mortgage lender doesn't take possession of the deed.
If you want a copy of your deed you can just go to the local land records office and get a copy of it. The lender then records a "satisfaction of mortgage" document in the land records office releasing the lien on the property. They are supposed to do that within 30 days of payoff.
The seller's attorney will give the original deed to the buyer's attorney at closing. That original then gets recorded at the clerk's office of the local municipality. The clerk's office scans and records the document into the land records and then sends it to the buyer or their attorney.