Let's just clear this up right now. Salt, as in sodium chloride, doesn't harden water. General hardness is from calcium and magnesium salts, and carbonate hardness is from bicarbonate and carbonate salts. Sodium chloride is irrelevant. You can add a bucketload of sodium chloride to a thimble of soft water, and it'll still have low levels of general and carbonate hardness.
Aquarium salt is indeed much like duct tape: covers over problems reasonably well, but doesn't fix them, and eventually fails, perhaps catastrophically. So I do like your analogy there. Sodium chloride reduces the toxicity of nitrite and nitrate, and this is probably why betta keepers are so attached to the stuff. If water quality is marginal, then sodium chloride might be enough to keep your betta healthy. But with the exception of a single, as-yet unnamed betta species (
Betta sp. Mahachai), all bettas come from freshwater rather than brackish water habitats. There's no reason to add salt to a properly run freshwater aquarium with adequate filtration and regular water changes.
If you need to harden your aquarium water, and there are plenty of good reasons to do this, then use a Rift Valley cichlid salt mix instead. There are home-brew recipes for this that use a tablespoon of Epsom salt, a teaspoon of baking soda, and a teaspoon of salt per 5 gallons. Or you can buy such a mix ready made. For most aquarium fish that don't specifically need hard water, like bettas, you'd use this at one-quarter to one-half the dose (i.e., as above, but per 20 or per 10 gallons). This will get the pH to the optimal 7.5 value where biological filtration works best, and the carbonate hardness will inhibit pH changes between water changes. Unless you have darn good reasons to do otherwise, soft, acidic water conditions are best avoided in most fish tanks, and especially ones as small as those most people keep their bettas in.
Cheers, Neale
IMO aquarium salt is fishkeeping's answer to duct tape. It fixes (or rather helps) pretty much anything! If your betta is healthy, however, I would avoid it because bettas prefer soft water and it will harden your water a bit.