The theory doesn’t seem sound when you give it more than a cursory glance.
For any cut or abrasion to heal, new tissue has to grow into the gap created. That new tissue grows as cells are brought to the damaged area in tissue fluid; the exudate from the wound. Providing an environment where that tissue fluid and those cells can do their job is the quickest way to promote healing, hence the use of dressings that keep the area moist and undisturbed.
Also included in that tissue fluid are white blood cells which mop up any bugs that are around, reducing infection.
Sluicing the area down with any fluid other than something neutral like water will only disturb the delicate process of healing, washing away the new cells and spoiling the local environment, so the process has to begin all over again.
Salt water may help to reduce the bacterial load, but in doing so it also messes up the healing process.
Tis old wives tale, nothing more, and that’s why we don’t do it in hospitals anymore.