I'm using PodcastRepublic on Android right now. It does a fantastic job of organizing my daily playlist for exactly what order I prefer to listen to episodes. The down side is that there is no easy way to translate this nice playlist stuff to the browser website. The state of the website is "mostly functional" and plays audio. Not much else. There is no sync to the Android app.

What I am going to try next is Audiobookshelf with a python script on their API to get the same playlist sorting features. I've got the architecture written out, but haven't gotten the time to write the code.

Reading into gpodder here is making want to give that a try, but the only website listed on this table doesn't say it syncs playback progress.

So what I'm looking for is something this can sort playlists like PodcastRepublic and sync playback progress like PocketCasts. AFAIK that combo doesn't exist right now.

Audiobookshelf

Any of these

https://podcasting2.org/apps

Antennapod on Android and Kasts on Linux, synced via Gpoddersync on Nextcloud.

I love Player.fm

They have both an android app with a lot of good features (stream or download, can set how many episodes to download at a time, when to delete old ones, all per podcast and as global defaults, set up your own categories/lists, display order, play order, etc) and an actual web site where you can log in and listen to your stuff as well with synced history, so you can pick up pretty much at the exact spot you paused a podcast on your phone (or vice versa).

I recently switched from Google Podcasts to Podcini (fork of Antennapod) and it's great so far

I personally like Podverse.

Check out Antennapod.

I personally switched my mom over to Pocket Casts after the news that Google Podcasts was shutting down. I don't listen to podcasts, tho, and I really just picked it because it was rated well. It's also not FOSS

I have Podcast Addict, I think its free but I paid a few dollars because of the great selection of settings and options I never knew I needed.

(For example I have it configured so that after 5pm the Sleeptimer automatically makes the podcast fade down and pausing after 10' of listening, unless I move/shake the phone during the fade down-part, then it extends for another 10'. Perfect for using it in public transport or in bed and dozing off.)