If you already have it, it looks like Plex can do it with https://channels1867.rssing.com/chan-55464362/all_p107.html
It'll probably get you most of those features, though it probably won't be as nice as something purpose built. But if you already have Plex it might be nice to have all your stuff in one place. Alternatively you could probably setup something to download podcasts to your server into a folder that Plex watches.
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.
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 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.
Have you managed to selfhost it ? Funkwhale looks great, but the installation process with another proxy than nginx in a container setup is far from ready and accessible to hobby selfhosters.
If you have no idea about proxies and headers forwarder... just don't waste your time and go straight to audiobookshelf !
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.