Vintage Ekco RCX 'Ekcosprite' - Volume won't turn down to zero

I was working on restoring an old Ekcosprite recently and came across a puzzling problem where the tubes all tested fine, as did all the components (many were replaced and new)... but I could not shut the voume fully off.  It took some experimenting and head-scratching - and finally a little Googling to figure it out.

Ekcosprite RCX