Why doesn't Apple TV on Mac recognise my Family Sharing subscription?

I have Family Sharing via my partner's Apple One account, but Apple TV on Mac doesn't recognise my subscription even though I'm logged in with the right Apple ID. (For reference: I don't have this problem on my iPhone.) I've tried restarting the app, logging out and back in -- all the standard troubleshooting steps, but nothing works. The only available options are to sign up for a free trial or subscribe -- which makes no sense considering my partner already pays for an Apple One Family Plan! How can I get Apple TV to recognise my subscription?