I agree, it "seems like that". But this a very frequent (even before Covid) question here in the forum. You only file KY.. You would only file WV if there had been WV state withholding, so you could get the withholding refunded.
"I know that I owe WV the state tax for the unemployment"
I don't know, for sure how it works in KY and WV, but the general rule is you report unemployment as home state income, not work state.
