Podcasts will not play on audio-only Connect devices because Spotify has labelled them as "mixed-media". They are not mixed media, they are audio files.
This is an issue that has been noted by the community for some time (around 18 months at least) and not addressed by Spotify in any serious way. It is not a niche problem as it will affect ALL users who listen to podcasts and own an audio-only Spotify Connect device. In other words, potentially millions of people globally. Please Spotify, rectify this issue as a matter of urgency. I for one am a regular podcast listener who bought an audio-only Connect device so that I could play podcasts through it, only to find that I cannot because of a categorisation error on your part.