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.