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.
To further elucidate the issue, and Spotify's lack of action on it, I point to an earlier (now closed) idea post submitted by another user (charango88) https://community.spotify.com/t5/Closed-Ideas/All-Platforms-Connect-Make-Podcasts-playable-on-audio-...
The advice given to this Spotify user was that this is "an issue that's worth investigating" and for them to find relevant help topics. However no relevant help topics exist for this problem.