In my car (2012 VW Passat), I have to switch the car's audio source manually to Media/Bluetooth and the Bluetooth option only becomes available when there is an audio actually playing. I have this problem all the time when switching from radio to Bluetooth for navigation, since the navigation plays audio only sporadically. I have to start playing some music on the phone first, then switch the car to Media/Bluetooth. I can then switch the music off: the car remains locked to Bluetooth, even if silent. Might be worth checking if your is a similar issue.

Regarding your search noise, no, I have never experienced it. I have it permanently on, visibility off.