You might like this website: https://alternativeto.net/software/ipod-iphone-podcast/?license=opensource&platform=windows
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I was surprised to see Pocket Casts on that list. I did not remember it being open source (though that might have been my standard when I was initially looking), and anyway I subscribe to PocketCasts+ , so it's a paid service for me. https://alternativeto.net/software/pocket-casts/about/
I like that playback resumes seamlessly when I move between my devices -- for instance between my car Android (which downloads files for playback later) to the office Windows PC (where I stream via the web app, which is pretty solid). Some of this functionality (but I forget which) comes with the PocketCasts+ subscription.
I have not used other podcast apps in many a year, so I don't know what the competition is like now. When I first started using PocketCasts, I found it by far the best option. Also, I have not used the Windows application.
Edit: the Pocket Casts website only lists the mobile apps as open source, and the Windows application is probably a web wrapper. Never mind.
Just out of curiosity, what's wrong with podcasts on the phone?
im going to give and answer and shill for portable apps for windows users. way easier to migrate machines when you just need to copy over a directory. https://portableapps.com/apps/internet/gpodder_portable
I've been using Podverse but I'm not sure if it meets your requirements. I just use it in the browser when I'm on Windows. The Android app doesn't seem like a web wrapper. Its source code is available under the AGPL. I've been paying $18 per year for the hosted service, but they provide instructions on self hosting.
Sadly we don't have huge choice. I think gpodder wirks on windows: https://gpodder.github.io/ and does the job.