Holy crap! This looks like it works. I'll see if phone audio stays on after I connect and disconnect one of my headsets. I don't know how you thought of doing this, but it's brilliant!This problem seemed to come up for me when the Google Assistant update hit my phone.
Try this.. It worked for me and my bluetooth works normally.
Go to the phone settings app>search for "Bluetooth scanning" and turn it OFF.
Turn on Bluetooth, go into your devices settings, and check if you can Toggle the "phone audio" profile.
Hopefully this works for you.