I watch (not on the Watch, ob!) and listen to podcasts, but haven't used Overcast. The Apple Podcast app is a big mess.
No podcasts are stored on my Watch. I don't know if Apple allows it, but there is some workaround to get a podcast playlist.
My major criteria for a podcast app is to be able to conveniently consecutively play one, two, or three episodes of the same podcast. That's it. Should be simple, but no. Apple's updated app adds two or three to do this. You have to make a play list, then it asks you 'are you sure'. Very counter productive workflow.
Previously you just set a timer. A poor methodology as you always end up with a podcast partially played. But it was fairly quick and easy, especially compared to the updated app.
I'd be all over an app, checking it out, if it would let me pick a podcast and then how many of the next eps I want to hear.