According to Spotify, they were not able to start developing an app until recently. They've finally released an app, but all it does is control the music playing from your phone.
I don’t particularly trust the rhetoric from either side on this issue. With regards to this point in particular, the API that they need to make an app was released earlier this year and many audio-playing apps are using it already.
The Overcast app can either control your phone or play podcasts that are "synced" to it, i.e., loaded onto the phone. At least from my reading of the WatchKit documentation, audio files playing from the watch must still be stored locally.
I don’t see how this is Apple’s fault, though: Spotify should be able to write an app for watchOS.