Thanks for trying Omar.
It still doesn't connect when I turn the radio back on (after starting the car, with the bluetooth device connected). Instead it tries to connect to Bluetooth Audio (BTA) for a fraction of a second and then defaults to line in.
I think I have a better handle on the problem now: When I typically leave the car, BTA is on and playing, I turn the ignition switch off (BTA continues playing), I open the door and BTA stops playing. I come back later, put the key in the ignition and start the car. For a brief second, the radio display actually says "Bluetooth Media Stream", but it doesn't wait for the phone to connect, it just quickly times out and defaults to some other source (usually line-in or USB (if I have a USB memory key inserted)).
So, the real problem is that the head unit doesn't wait long enough (it takes more than a fraction of a second) for the bluetooth device to reconnect before it the head unit goes looking for another audio source.
My conclusion: This is a software problem. The software is telling the head unit "if you don't find a connected bluetooth device upon startup, then find some other audio source to play". I assume this could be fixed if Ford cared, but after reading hundreds of blog posts on this (and other) radio related issues, it is clear that Ford either doesn't care or doesn't know how to fix the problem(s).
If they'd give me the code, I'd find a fix for it, but I'm sure they wouldn't do that. Actually, if they would provide the open source software community with the code for all their "smart" radios, I'm sure they would end up with a much better product. But that will never happen either.
I guess I am forever doomed to saying "Bluetooth Audio" every time I enter my 2012 Ford Edge.
How ridiculous!