As for the "Sound Connect App" that's unfortunately the core of the problem. That app doesn't exist for Linux. If the hardware relies on that app to set up or manage profiles, it creates an unavoidable roadblock for desktop Linux users.

The app runs on your phone (Android or iOS), and then you use the phone to manage Bluetooth connections for the earbuds. IMO you shouldn't need a second device, but I guess they just assume 99% of people are connecting to a smartphone.

It just seems to be a non-standard implementation from Sony that doesn't play well with the standard Linux audio stack.

I think the issue is that the actual Bluetooth connection is obfuscated behind a proprietary connection to the app, and the app exposes the protocol.

I agree it's a stupid implementation, prioritizing a UI for pairing over literally everything else, but you still might be able to get it to work. I've successfully paired my WF-1000XM4 earbuds with my EndeavourOS (KDE) desktop.

I have a pair of Sony WH-1000XM6s (similar sort of pairing and Bluetooth system although with the addition of LDAC) and I can confirm the Sound Connect app is not required to pair the headphones (works perfectly on my laptop with a Qualcomm Bluetooth chip and my desktop with an Intel chip, both running Fedora). It's only for settings management and firmware updates through your phone. It might be worth checking if there's any firmware updates for the headphones, and also try a few more resets. I used to have a pair of Bose headphones that had the exact same problem as you are experiencing, but it would go away after disconnecting and reconnecting a few times. It's worth also seeing if you can pair with any other type of device in addition to your phone (Windows, smart TV, anything that can do Bluetooth A2DP) as it might be an issue with pairing to devices in general.

Out of curiosity how are you triggering pairing mode? If you are triggering pairing mode through the Sound Connect app, it might be worth instead holding the hardware button on the back of the case for 5 seconds to enter pairing mode.