Update to my previous replies.
TL;DR: I purchased a new 2025 car and bluetooth now works flawlessly with it. Makes me think the issue is something about newer bluetooth protocol changes and lack of backward compatibility in the iPhone 16 software.
Summary: when I upgraded from an iPhone 12 to a 16 PRO, bluetooth would not work in my 2019 vehicle. It also wouldn't work with another device (a pickleball ball machine). But it did work with my (old) Apple ear buds.
I tried all the suggestions here, from forgetting all devices (and personal data) in the car to completely rebuilding the phone from scratch (which was done by Apple techs in an Apple Store) and that didn't fix the problem.
Now it works in my 2025 vehicle. I assume it still doesn't work with the pickleball machine but I haven't had a chance to try that (I don't own it).