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.
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
Just out of curiosity, what's wrong with podcasts on the phone?
Sadly we don't have huge choice. I think gpodder wirks on windows: https://gpodder.github.io/ and does the job.
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.