Playlists are not visible any more in the podcasts app. Apple does not provide this feature.
When you are using the podcasts app, podcasts are disabled in the music and video app. You will only see podcasts in the music app, if the podcasts app is not installed. If you don't need 2x in video, and you liked your workflow in the time before the podcast app, simply remove this app.
thanks. I uninstalled the podcast app, and now everything is in my music app, including playlists containing podcast episodes. In the process I learnt how to create smart playlists to save space on my phone. Disappointing that the podcast app was created, released and installed automatically with the OS without thinking through all the use cases of how people use podcasts or informing users and giving us options to choose what we want to use and how.
I hope apple will improve the podcast app soon.
The only thing the podcast app does better than the music and movies app is, that you can play video podcasts in 1 1/2x or 2x.
Another missing thing in the new app is, that the podcast albums are not synced well. I also copy videos manually to podcasts in iTunes and this manually created albums only displayed in one album.