Have you attempted to reset the network settings on your device? Just a heads up, this will delete any previous devices that have been paired to the device. You will have to re-add them back.
I ran into the exact same issue w/ my new S20 FE 5G that was purchased this past weekend.
This webpage's directions resolved the issue for me --
At first, it connected to one of my headsets and was working just fine. But at some point the sound dropped out and changing the volume level didn't do anything. I had the phone forget the pairing and tried to re-pair. At first in the Bluetooth menu, after pressing Scan, it would "see" my headset, but it wouldn't allow me to connect to it. I tried it multiple times to no avail. I gave it a try again the next day, but then it wouldn't even see the device after scanning. I confirmed that there wasn't anything wrong w/ the headset by connecting it to another device.
I then tried a different device and the exact same thing occured; phone didn't recognize it and wouldn't even allow me to pair. I tried turning BT off and on again, and the same w/ the BT device. Nothing worked.
However, after following the directions linked above and clearing the cache (and data) of the system Bluetooth app, the bluetooth menu was able to scan properly and recognize devices for pairing.
(One other thing to note is that I had used the Samsung SmartSwitch app to transfer all my settings/configs from my prior phone; this included all of my existing BT pairings/connections. Both of the devices that I tried to connect w/ were brought over were "old" connections from my old phone. I wonder if there's something buggy about bringing the memory of those connections over from the old phone perhaps?)