I was so frustrated by Apple’s dropping of iPod Shuffle support in the Podcast app of Catalina! I have a few 2nd gen shuffles and use them for podcasts. Like others have mentioned I prefer the portability of the Shuffle when I’m out and about to my iPhone or older iPod.
Maybe this has been mentioned in another thread but I heard about the free app Retroactive From GitHub https://github.com/cormiertyshawn895/Retroactive/blob/master/README.md which allows installation of iTunes again, and downloaded it, followed instructions, and presto, I have iTunes back and all my podcasts are there, as well as iPod Shuffle support as before!
So far I’ve had no issues whatsoever on MBP running Catalina, but I no longer have any need for its unwieldy and unsupportive Podcast app. I hope my experience helps others.