As an avid podcast listener I use Pocket Casts. But I have been looking for something to replace this, especially to own my listening status and save favourite episodes.

Either I haven't tried it properly enough or there simply isn't anything out there.

My experiences:

Podfetch: Very very basic, can't get Favourites to work, doesn't save specific episodes into a permanent archive, couldn't get playing to work, no mobile apps

Plex: Cancelled Podcast support two years ago

Audiobookshelf: As described by others, performance issues (I have ~117 podcasts and updating takes ages), iOS app is Testflight only and BETA is full. Wanted retest with it being only an intermediary.