Partial solution.
I got this working in Google Maps, but Waze is still not integrating directly with Driving Mode.
This is what I did:
Open Google App > Settings > Google Assistant > Transportation > (scroll to bottom) > Driving Mode > Enable When Navigating in Google Maps
Edit: I noticed the "When navigating in Google Maps" option is disabled in my stock phone... So strange.
Note that I have 'Personal Results' turned off. This setting doesn't seem to matter. On my stock phone, I also have Personalization turned off and Driving Mode in Waze works fine.
It's still necessary to manually touch the Microphone icon, but "What's my ETA" and such works in Google Maps. It still does not work in Waze. GRRR!!