Great hint with the LineageOS App 
it seems to work well 
Can I trust that the LineageOS phone app not to share my calls and contacts with google as both have the google privacy policy?
(I am not familiar with LineageOS so far).
I tried to replace the Google Phone app before, too.
Ended up with using Simple Contacts (F-Droid) as default phone app, and Open Contacts (F-Droid) to show call log and in-call notification (overlay with name).
My observations:
- Even with Simple Contacts as default phone app the screen during outgoing and incoming calls looks the same as before.
- With deactivated Google Phone app no screen at all apears to manage active calls.
Because of this I wonder whether Google Phone and LineageOS Phone using a different UI of the same system app like Call Management for example (similar logo)?
I am not sure which app @corvuscorax meant with system Dialer app?