Speech output is enabled for Google now but I don't have that BT option. My car only supports phone calls, not media (like playing music through BT).
To fix the broken setting that causes Google search / voice to text apps to not work with Bluetooth Microphones, do the following:
Note: This will erase your google data on the phone maybe including your linked credit card authorization from your bank in android pay (tap payment only, not Wallet.) which I seem to recall having to re-set it up with my bank since they are very secure with that, so make sure you are prepared for that possiblity. It won't delete your linked accounts or authenticator two factor auth. Just the core googly stuff and settings related to the local settings on the phone.
Go into settings > apps > Google. Disable it to reset it back to the original version, then enable it again. Still in the Google app, click on storage, then manage space > clear all data.
Go back to settings > apps > show system apps. Find Google Services Framework. Click storage, click clear data.
Your Bluetooth microphone should now start working again. I have had to do this on two phones so far. You may just get away with deleting the data on the framework app, I usually do both since I don't want to mess around any longer than I have to.