AirPlay turned off and can't be turned on

I have a MacBook Air and with no apparent reason the AirPlay icon has dissapeared from the desktop. When I go to System Preferences → Displays, the status of AirPlay shows "Off" and there's no way to change it since it doesn't show the "On" option.

The AirPlay icon in the menu bar is only shown when an AirPlay device is available, per the option ‘Show mirroring options in the menu bar when available’ (emphasis mine).

You're unable to change the option from Off because there are no AirPlay devices to choose from. There is no ‘On’ option ever, only a list of devices to choose, except there are no devices found so the list is empty.

There is never an AirPlay icon on the desktop.

I ran into this same issue and tried a number of things suggested elsewhere; the only thing that worked for me was the Floris' solution, which is to sign into my iCloud account. After that, the "on" option appeared instantly and I was able to connect to my iPad via WiFi and USB.

Running BigSur 11.2.3 on MacBook Pro (16-inch, 2019), connecting to first-gen iPad Pro.