Update:
I have noticed that this problem really only occurs at my house. I have fully reset the phone with no success. I currently only have my Powerbeats Pros, Garmin watch, and Garmin bike computer paired with my phone. When I am away from my house, the Powerbeats Pro play just fine. When I am home, the iPhone stutters and skips when playing any music, media, and on phone calls. I can play from my iPad or any of my Windows computers or my Garmin watch perfectly. It is like the iPhone is experiencing interference that no other device is experiencing. This issue is new from about 13.5.1. I am wondering if the iPhone could, in some fashion, be trying to connect to other Bluetooth devices even though it is not paired with them. I did a scan of the local area for both WiFi and Bluetooth and both are quite busy. I played with my local WiFi setttings - changing the 2.4ghz channel, reducing 2.4ghz power, disabling/re-enabling 20/40mhz coexistence, etc.
This problem occurs whether I am playing downloaded or streaming content. The problem occurs regardless of my location in my home (3,100sq ft).
I have turned off all Bluetooth access in the privacy settings and that did not help the problem.
I have turned off WiFi and cellular data and neither helps the issue.
Given that every device besides the iPhone works perfectly over the same Bluetooth headsets and that the iPhone works perfectly away from my house, I really believe that this is a software/firmware issue with how the iPhone is handling Bluetooth traffic while in a busy environment.
Does anyone have any thoughts on fixes or are we beholden to Apple fixing this?
I have tried upgrading to 13.6 Beta 3 and that did not help either. From what I can tell, Bluetooth modem firmware remains 1.06.00 in 13.6 Beta 3, which is the same that was at 13.5.1.