Salt adds electrolytes, which reduce osmotic stress to the gills. This is especially important during disease, when the ability of the fish to maintain homeostasis with its surrounding water is disturbed.
Salt can aid in the production of the slime layer and speed up the healing of some wound sites. This occurs by hyperosmolarity... fish have a certain specific gravity (concentration of total dissolved solids) in their bodily fluids, and so does the surrounding water... if the surrounding water has a slightly higher concentration, the fluids from the wound site move into the water around it, and fresh plasma goes in to replace it, creating more blood flow in that area (makes it heal faster).
Many pathogens, such as Ichthyophthirius (the protozoan that causes "ick"), do not particularly like salt.
It is true tetras don't need a lot of salt, but they still need a little. It is also true that it can do more harm than good for beginners, but if carefully done it can be beneficial
"Prevents ich"
The concentration needed to have an effect on the parasite is much more than would be healthy to maintain at all times in an aquarium - especially a tank full of fish hailing from soft, acidic environments. Many people add salt to african cichlid tanks, because Africans are from areas with very high TDS in the water. (Mind you that african cichlid salt supplements contain more than pure NaCl) However, many fish such as tetras, rasbora, angelfish, discus, rainbows, etc. come from areas with low TDS, meaning they are not used to having this in their water. Their salt tolerance is much lower. Not to mention that keeping the "ich fighting" concentration of salt long-term would be detrimental to a planted tank.
"Adds electrolytes"
Salt does add electrolytes, yes, however the amount needed in your water is miniscule - so small in fact, that it is in the tap already.
"Aids in slime layer production"
Yes it does. And, for that matter, so do many waterborne irritants and toxins. Slime coat is created by specialized cells scattered throughout the epidermis - they can be stimulated to grow by either a hormone or an irritant. NaCl has no known hormonal properties, thus it stimulates those cells by stressing the fish.
"Helps heal quicker"
In a dry environment, salt helps draw degraded cells and pus from a wound, thus helping clean it. This concept, however, cannot be extended to an aquatic environment. The idea that salt helps fish heal faster is, as you said, based on hyperosmolarity - the tendency of a more concentrated solution to draw a less concentrated one. You cannot do this with a fish - a brine that is more concentrated than the fish's own blood and tissue will kill it in a very timely manner.
In the end, the only good use I have found for aquarium salt is treating ich. In the meantime, put the salt in a fancy dish and set it on the table - sprinkle it on some eggs - it's just NaCl after all, the same as table salt - guests will be impressed at how much you paid for your fancy salt
