I wonder what the copywriters meant by "gummy" edge? Then they recommend STONOIL, which is likely made by the hone manufacturer. The STONOIL is then declared to be "non-gumming." So gummy is good on the edge but bad on the hone. If lather was possibly the most popular topping for the barber's hone at the time, the manufacturer may have just been trying to peddle an auxiliary product, a fairly common sales strategy.
A "gummy" edge might be the TPT which is more sticky than gummy. In some contexts, the two words could be almost interchangeable.