Is the operation of a hard-start domestic water pump improved by the addition of an external soft-start controller?
I would say a soft start controller would not change the problem I have many booster pumps and the only thing a soft start would be good for would be throttling the pump. Centrifugal pumps create a large hammer action in how they work.
A expansion tank may be your best option. The quality of the valve may also be the issue I can warn you I had a irrigation pump that when throttled to 45hz with a VFD (can be used as a soft start) that frequency would tear up 4” irrigation lines I could run fine from 50 up to 70 hz and had no problems (below 40hz the motor would over heat)
There is cavitation at start up on the pump this is normal but a soft start won’t stop that. Some water filters can not be used without a pressure tank or expansion tank as the vibrations tear them apart.
I did see the comment on the water hammer arrestor that may work but I would think a 2-3 gallon expansion/pressure tank would solve the issue better than a electronic method.
I thought I should add most 120/240v pumps are capacitor start with a centrifugal clutch pulling them in. A soft start in this case would probably overheat the motor they are designed to spin up quickly and change windings or the cap depending on the type.