Don't know what the procedure that the bike shop is intending or what solvent they use, but may be adequate.
Also, don't know what the WD40 version you tried. People here have used normal WD40, which has a large solvent content. Maybe repeated attempts would improve your results?
The Shimano recommended method for maintenance is shown on their tech website. In summary, it says to remove the shift unit and rinse in kerosene. This does require resetting the brake action return spring, which is tricky at first. The spring is easier to set with the shown tool but a sports ball inflation needle may be used or a split/tension pin of the correct size.
Here's part of the Shimano procedure.
Shimano's directions also shows removal of the metal bracket that the brakes cable hooks into but is not shown above since most people don't have the special tool to undo the funny shaped (castle shape) nut.
Detail of the tool to help install the brake spring and substitutes.
Shimano tool on top, ball inflation needle, split pin, 2 split pins pushed together (split pins are available in hardware stores).
The most complete way to refurbish would be to do a partial disassemble of the whole shift unit as pictured in my prior post (1st picture) where you have access to all rachets and pawls. This does take some knowledge and can be tricky since a few other springs need resetting.