In one word, yes. Your mother and grandmother were right when they told you to gargle with salt water when your mouth hurts.
Using warm salt water does help the mouth recover after surgery, and Dr. Schafer does recommend it. A study in 2014 showed that patients who had their wisdom teeth removed and who rinsed at least two times per day after surgery were less likely to have complications and post operative pain. The use of salt water has been documented by ancient Egyptians as well as ancient Greeks. It is great because it is cheap and safe.
Now generally it's best to gargle with salt water the day following the surgery so that the clot that has formed around the surgical site can set and heal. Make the water super salty and not boiling hot but warm enough to be slightly uncomfortable.