In an interesting twist, BeyondPod, a podcast manager that has been around since seemingly the beginning of Android, has been removed from Android Auto. This isn't entirely surprising - prior to this April, it hadn't had a public update in two years, and people had been complaining about the huge amount of bugs.
BeyondPod was once one of the greatest podcast managers out for Android, and it still has its fans. But its beta had been buggy for months as of January 2018, and the public release three months later only spread the problems. That's obviously not something Google would want on a platform such as Android Auto (even if it still has its own problems to work out).
BeyondPod users took to the Android Auto User Community to see whether the podcast app had disappeared for others on AA. Shortly after, members of the Android Auto team confirmed that the app had been disabled, primarily due to the fact that BeyondPod was stealing focus from other media apps.
However, not all has been lost; Google is in contact with the BeyondPod team, and when a fix for this is ready, it'll be tested, pushed to the Play Store, and reenabled for Android Auto. Google was unable to provide an ETA. In the meantime, you'll just have to sit tight.