I have heard “chops” as in “skills with an instrument” for as long as I can remember. That’s at least as far back as the early-mid 1960’s.
I never thought or considered it related to “axe” as in “chopping with an axe.” Even though “axe” does usually mean instrument.
I always thought it was more like “baring one’s teeth.” I have heard “He’s got a good set of chops.”
There is also “bust his chops” meaning, (I thought) to figuratively or even literally strike in the mouth.
As for the OP: “not much chop"--I do not recall hearing it. Also, I do not recall hearing “cop” used interchangeably with “chop.”
I personally sense there to be a distinct difference in the two phrases: “Cop some wine” and “Chop some wine,” with “cop” being more like “to acquire,” and “chop” being more like “to take,” “to steal,” or “to rip-off.”
This is anecdotal, from personal experience.