Feature you want:
Control AntennaPod from Android Auto.
I keep seeing how the Google Poscasts app is handling this and it’s a completely different experience.
AntennaPod should already support Android Auto and the technical details you linked are implemented. I do not use a car, so I do not have experience with Android Auto, though.
I just checked: The Google Play version of AntennaPod is approved by Google for Android Auto. Apparently, there is a developer mode for Android Auto that allows to enable unknown sources. F-Droid is signed with a different key, so it is probably not approved by default.
Alternatively, you can switch to the Google Play version
I did! I still need to figure out how to make it work directly, at first the app is unavailable and when I start playing on my phone it becomes available.