Operation without power: Passing analog video
You could use a depletion-mode FET like CPC5602 or DN3135.  But the obvious way to connect it would require the use of a negative gate voltage to open the circuit.  The negative voltage doesn't have to supply much current however, so you can probably arrange a simple charge pump with a square wave and a couple diodes and caps.   When the square wave stops, the transistor will turn on.