I have the same issue since I upgraded to Monterey (couple hours ago). Bluetooth won't work and I am working from home due to COVID-19 Pandemics. The lack of Bluetooth makes it harder for me to work since I have many devices connected such as Mouse, Keyboard and earphones to enter calls that get noisy with built-in mic or speakers.
I've had this notification of "system update" on my Mac for a while and chose to ignore it. And now that I finally installed it, it is buggy. Apple should test its software in all supported devices because I wouldn't install Monterey if it was not supported.
I have tried the following commands that I found in the internet already:
sudo kill -9 $(pgrep bluetoothd)
sudo pkill bluetoothd
The first command makes it possible for me to turn the bluetooth on but no devices are shown and in the System Report, I still see:
Discoverable: Off
I guess, I will try to buy an external Bluetooth hardware tomorrow even though I know that this is a software-related issue, I cannot wait until Apple decides to fix this for Retina MacBooks, its not their most recent device, so I guess no priority here.