I have noticed that you can control the Spotify on car playback via your phone app, provided you tell it to play back on the Tesla via the app devices menu. So you could go that route. It was acting a bit wonky though and the app kept defaulting back to playback on my phone whenever I would wake up the phone.
I haven't found a way to access podcasts through Spotify (though you can access many through TuneIn). But the global search in the Music tab (magnifier icon) returns results from Spotify if you are signed in. Or you can use a voice command.
So far only way I've seen to access podcasts via Spotify is to add them to a playlist, since a Spotify playlist can include must or podcasts. I don't know of anyway though to have a new podcast auto added to a playlist so have to manually app it to the playlist each time.