True, and actually some cleverly designed BLE devices advertise only for a limited time after pushing a button. I don't see why classic BT devices shouldn't do

True, and actually some cleverly designed BLE devices advertise only for a limited time after pushing a button. I don't see why classic BT devices shouldn't do the same and work (almost) like you described: only advertise for a minute or two when asked to, and shut up for the rest of the time.