Spiritual successor to podgrab is podfetch https://github.com/SamTV12345/PodFetch

Something I have found is missing from both of these suggestions as well as every podcast app on device is transcoding to speed up so it is not sped up on the fly. For a lot of phones and other devices the task of playing back at 2x speed is enough to demand a higher power state than what is required to play a sped up file. For efficiency doing a single pass of speeding up the audio then playing back at that speed would use less power during the playback phase, allowing you to download and speed up all of your podcasts at home while on charge then listen for long periods without completely killing the battery. I have checked with a few if the open source devs and this is not a feature they see utility for so nobody intends to make it.

I’m ever continuing development on my self hosted podcast management app with all the features you’re looking for here https://github.com/madeofpendletonwool/PinePods

It’s not in a finished state yet. But it’s definitely usable as is. It’s fully dockerized and hosted on your server, and streams from your server to clients.

I use Gpoddersync on Nextcloud as the subscription/watched service, Kasts as the desktop app and Antennapod for Android. I think there’s a Podder webapp for Nextcloud that uses the sync data as well, but I’ve never used it.

https://github.com/joshwbrick/podsumer?ref=selfh.st was featured on selfh.st this week. Can’t say I know a great deal about it, but might be worth checking out.

I have no idea about podcasts. But since no one mentioned it yet, perhaps take a look at Funkwhale. It’s a decentralised platform, like Lemmy, for music and podcasts.