There must have been some talk about this but I was not able to find it here or elsewhere.
The current behaviour (in my case 1:41.7-0ubuntu0.22.04.8) seems to be that, whenever the bluetooth tab is selected on the gnome-control-center, scanning begins and does not stop until tab is changed or the app is closed. This is not too bad, but probably you do not want or need this to happen if you are not setting up a new device, i.e. most of the times you would open it.
I have recently had some trouble with bluetooth devices, trying to connect them on a new computer. I finally solved them but In any case, I have become slightly familiar with the bluetooth daemon CLI bluetoothctl. Knowing its internal commands made me think it would be nice if the control center could toggle scanning on and off. For instance, if we establish a parallelism between the GUI and the CLI, opening the window would call “devices” to list those already paired while toggling scan on would call (you guessed) “scan on/off”.
Of course I know there might be a reason for this. e.g. KDE’s bluedevil has this button, yet it has more problems connecting, but I thought it would be good to mention this. I must also note that sometimes devices appear twice with the same or different name, and one of them MAC addresses would be connectable while the other not (so maybe the behaviour is intended).