Still haven’t found anything dispositive, but I found another tidbit. The 1913 publication, the century dictionary and cyclopedia, defines Chicago as to skunk or whitewash another player in cards or other games. And it even suggests that the slang term Chicago came from the idea that the city’s name was “assumed” to come from the Algonquin word for skunk (suggesting that that might not be how the city got its name, but that the idea that it did is where the term came from.). 1913 is too late for this to reveal whether the cards, baseball, general, or political sense was the “original” one, but it suggests that the terms are linked and that the cards sense was first.
The 1897 edition gives a similar definition of “skunk”, without a Chicago synonym, and references “cards or billiards”.
A 1880 printing of the British farmer’s magazine tells an anecdote, “a matrimonial game of euchre” where a husband and wife tease each other playfully while playing cards. The wife invites (dares) her husband to make a certain play, and he protests that he has already scored two points, and if he plays as his wife suggests he will skunk her. She replies that she would like to see him try to score a point if he did so. In this context, skunk seems to clearly mean shut out, and it is used in the context of a card game.
The 1864 publication, the American Hoyle, defines a slam as a love-game, and notes that a vulgar name for this is skunk. Assuming “love” here means what it means in tennis, this would again suggest that skunk meant shutting the other side out. And some contemporaneous definitions of “slam” assign it the same meaning, I.e., to stop the other player from scoring.
Some publications of about the same age refer to “whitewash” as having the same meaning as skunk has, and also apply it primarily to cards, but without a reference to Chicago.