I've been searching for an alternative to Android Auto itself. After an update some months ago, launching the app takes me to a settings panel declaring that Android Auto is no longer supported for phone screens (i.e. it is now only designed to run while linked to a car with a compatible display), and the app itself refuses to run as it used to on just the phone.

Add that functionality to the Google graveyard, I guess...

Possible workaround: downgrade to an older version and disable auto-update?

I don't think Android Auto works without Google Play Services at this time. Here's the feature request for Android Auto support on microG:

https://github.com/microg/GmsCore/issues/897

Lineage works perfectly fine for me with Android Auto

Can you please share your specifics? Phone model, LineageOS version, Android Auto version if you can?

It will work if you install a Google Apps package, but not otherwise.