I found the answer on a Subaru owners' forum, so hopefully if anyone else is having the same issue with a similar car this will work for them too. The stock stereo unit in my car (2013 Impreza WRX) has two different methods of pairing the phone, one via the front panel controls on the stereo, and another through the hands-free voice commands. Only pairing via the voice commands will let the phone auto-connect to the car. Unfortunately after fixing that, a different problem that existed with my other phone still exists - the bluetooth audio sometimes drops out entirely even while the phone is still connected and requires a bluetooth restart/reconnect on the phone. Given that this problem happened with two different phones I'm guessing that problem is with the stereo
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