limescale flakes in shower pipes
A standard water softener will remove most of the calcium ( and Mg, Ba, Sr , etc) from the water supply. It will not remove existing scale in the pipes, etc. It would be expensive to remove the existing scales ; flushing with dilute hydrochloric/muriatic would do it , but not a job for the homeowner. The taps that don't plug may have no strainers; also you may have mostly "temporary" hardness. In temporary hardness , carbon dioxide acidifies the water and keeps the calcium in solution, when the water is heated the CO2 is removed, and then the calcium precipitates/scales.