Thanks for the reply. The steps at that link seem to be for people who can't get their Bluetooth devices connected. I have no problem getting connected. And in fact I just did a fresh connection to a new vehicle with the same results.
However, after much Googling on this subject I think I have found the answer, and it's not a very pleasant answer at that. Apparently with Android Pie, Google, in their infinite wisdom, decided that ALL notifications and sounds MUST go through the Bluetooth device, and not through the phone's speaker, if the phone is connected to a Bluetooth device. Most people are complaining that this means that the text notifications come through their Bluetooth headsets and not the phone, but in my case, for some reason, the text notifications aren't coming through my Bluetooth device (the vehicle's audio system).
So, apparently my phone is working pretty much as Google wants it to. No longer sending text notifications through the phone's speaker when it is connected to my vehicle's Bluetooth hands free system. The only question left is "why doesn't the text notification come through the vehicle's Bluetooth hands free system?".
I did find a workaround in my Googling. In the Bluetooth settings (when connected to the vehicle), if I turn off "Media" the text notifications now come through the phone's speaker again. Of course this means that I can't stream media through the vehicle's Bluetooth hands free system, but I seldom do that, so turning off "Media" is a trade-off that I can deal with. When I do need to stream media, I can turn it back on again.
So... I guess my new question is "Any idea why the text notification doesn't come through my vehicle's Bluetooth hands free system (the vehicle's speakers)?".