I also have been using podcast addict, probably for over 10 years now. You can autodownload new episodes (mine checks daily at 7a and 7p), create custom playlists for juggling multiple podcasts at once, has android auto support, and you don’t have to deal with the bs of Spotify.
I’m sure there are a 100 other features I don’t know about, but that’s what I use.
It’s not strictly privacy-focused but The Goblin’s Notebook is designed exactly for your use-case. It has markdown, object connections, every object has a player visible setting, so your players can access known content while you keep secrets hidden. There’s a free tier, a mid tier at $1.50 and an unlimited tier at $3 dollars a month (managed via their patreon).