I have a Verizon S3 and a '10 Jetta TDI with factory bluetooth (phone only, no audio streaming) and I have replicated that if your screen is on then you auto connect, if it is off then I have to open the bluetooth settings and connect manually.
But a problem I have found that drives me even more nuts... When I am connected to my cars bluetooth and I have an aux cable plugged into the S3 to listen to podcasts, etc. If a call comes in and I am not listening to anything on the S3 then when I answer a call it tries to go thru the aux cable and the toggle button on the call screen is greyed out - I have to pull the cable out then BT kicks in! If I am listening to a podcast or music then the BT works fine.
Definitely a bug. I am going to call VZW today about it.