First off salt is salt (NaCl) you can use any kind (aquarium salt is BS), as long as its just NaCl, I use water softener salt to treat, a 50lb bag for $5.
When a white spot leaves a fish, the spot is 100 or more new ich being hatched, it doesn't mean the ich is dead when it leaves the fish, it just means there are more.
And the new ich are microscopic, you can't see them attaching to the inner gill membrane with the naked eye.
Unless you kill those new ich that fall (hatch) off, it prolongs the disease.
Its when the ich comes off, that it is when it is vulnerable. It is killed by heat, or salt, or meds. The meds changes the osmotic conditions in the water, so the ich basically implode. It takes time to eradicate ich from the tank, you must kill every emerging parasite, it helps to heat the tank, because beside killing new ich, it speeds the life cycle. Vacuuming also helps, because some dormant ich can sit in the substrate for different lengths of time. It is programmed to do this, in case conditions at the moment aren't right, this way, in case they aren't right, some survive.