This has been a recurring issue for me across two different Apple TV's (HD and now 4k) over the last fourish years I've had one. It's infuriating. Sometimes I can quit out of the app and restart it and it'll update but sometimes it just refuses to do so, even when it fully updates on iOS. And the syncing between iOS/tvOS/MacOS is atrocious. I constantly get played episodes appearing on the TV and even after clicking "Mark as Played/Unplayed" several times, they'll just stick around for some reason.
It's very annoying because I primarily use my AirPods through the TV and hate having to switch to the phone solely so I can listen to podcast episodes that I KNOW have updated. Oh and it also constantly keeps episodes downloaded even with "remove after played" and desyncs settings between devices so that I constantly have to go back and change them all. It seems like such a poorly maintained app, as if Apple has just said, "Eh, good enough," and then forgotten about it.