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I am with you and suspect that a lot of this is just being fussy but not all.

A good shave brush will last decades. Simply using it keeps it clean
for me.

.....But like kitchen sponges --...

Additional info....
"Borax, also known as sodium borate, sodium tetraborate, or disodium tetraborate, is an important boron compound,"

So while not calcium bicarbonate the point is correct that...

Pay attention to your local water.

Like anything in the bathroom you will get soap scum
build up and depending on your water you may also
get mineral build up. i.e. shave soap is soap. Minerals...