I somehow ended up on the BrcmPatchRAM.kext release page again and saw the change log for version 2.7.0 saying, "Improved performance on macOS 15+ by disabling incompatible patches". That got me wondering if my card was affected. I replaced BrcmPatchRam.kext with version 2.6.9 and reset NVRAM, and my Bluetooth back. Not too sure if it’s supposed to show up as a third-party dongle, though, but at least it works.
AirDrop remains limited as it started from Monterey: it only works reliably from my iPhone to my Mac. If I want to send something from my Mac to my iPhone, I actually need another iPhone nearby, set to discover others for AirDrop to work both ways.