Kind of what I was thinking. The only source the radio would offer that isn't available on the phone is AM/FM, which depending on the phone may actually be possible though its rare these days as it generally relies on the wiring in a set of headphones to serve as the antenna.
I've not seen a vehicle that transmits its own audio via Bluetooth, its always a receiver.
If you truly want to get radio output to Bluetooth you can install a basic line output converter to factory speaker connections, as if you were installing an amp and pick up a Bluetooth transmitter from whichever vendor you prefer. This would output anything playing to the adapter.
Whole thing would probably run $50 and is fairly basic wiring. The two I've used in the past had Rx and Tx modes and accepted RCA input through a 3.5mm adapter which was included.
