Very interesting concept that I never considered. Shower with salt water and then rinse with regular water. Pay one price for salt, one price for regular water.
The salt water would probably be too corrosive to pipe significant distances, which would make that difficult to get enough scale to make it feasible.
Most of the dual-use water systems I’ve read about were fresh water vs gray water.
Much piping these days (>75% for major distribution pipes, and increasingly common in residential use) is PVC, which doesn't corrode.
Salt and minerals get into fittings, destroy valves, and coat everything when the water evaporates. Probably not a great solution.
You would have to purge your system and shower head with vinegar every day just to keep things clean.
You are assuming that only metal pipes exist. Nowadays most pipes are made by either concrete for the big ones (canal like) or plastic connected with glue that is rated for hot or cold water.