There are two things to address in your question: Whether Beeminding podcasts is worthwhile, and how to do it efficiently. I can speak to the first:
December of 2014 into January of 2015, I used Beeminder to nearly halve my backlog of unlistened-to podcast episodes. My motivation was to be more caught up with current events. It worked well for me, and was one of the more fun things I’ve Beeminded, since I enjoy podcasts so much — “Sorry, honey. I’ve got to do this fun thing right now or I’ll be fined!”
I just entered data manually, though, so I don’t have any experience with integrations. (And for some reason I didn’t archive that goal long after I considered it hit, which is why there’s that long flat tail of no data.)
FWIW I use the Downcast app for iOS. I’ve enjoyed it for years, but recently it’s suffered from a growing number of bugs and seems like it’s being neglected. As far as I know it only stores data locally on my phone; it doesn’t sync to any service that could be used for an integration.