As an air traffic controller, for 20 years, I found if a pilot turned off his transponder, we got no altitude readout at all, and it was extremely hard to identify an aircraft.
You are obviously much more aware of how this all works on a day to day basis that I am. And obviously with ADS-B (Automatic Dependent Surveillance Broadcast) the whole game changed. So yes, without a transponder airplanes are just a blip on an air controller's screen. And without the pilot or someone else communicating the position of the aircraft the controller has no way to identify the blip unless it was tracked from takeoff. However, military radar systems can track aircraft and missiles without transponders and determine their altitude.
I assume that this area is under surveillance by military radar systems where the turning off the transponder would not necessarily make much of a difference. But then it all comes down to who is watching, what they are looking for and what their motivations are.