Turns out that my motherboard's WiFi antenna is necessary for Bluetooth to work correctly
I was troubleshooting Bluetooth connection today and I thought that this is somehow related to software (PipeWire, bluez, bluetoothd and all that stuff). But no. Apparently Bluetooth barely works when WiFi antenna is disconnected from my ASUS motherboard.
Anyway, this might save a lot of time for someone, so I'm posting it here.
I had the same problem too, I ended up buying a USB Bluetooth transmitter. I figured the board was bad until I replaced it and had the same problem with the new one