This is the closest to the correct explanation. The reason televisions based on AM radio reception showed static is because of a circuit called the AGC (Automatic Gain Control) which worked like a robotic volume control. Its job is to keep the recovered video signal within a certain amplification range. As long as there was a carrier (the TV station was “on the air”), you’d see whatever the station broadcast. But when they turned off their transmitter, the signal strength would fall and the AGC would increase the amplification until what you see is white noise, mostly due to the random motion of electrons in the electronic components. We can minimize that by cooling, but it can’t be totally eliminated. Audio amplifiers often come with a “hiss” specification that tells you how much of this kind of noise you can expect at normal operating temperature.

BTW, modern digital TVs -will- show a noise picture if they lack a video muting function when no carrier is detected. I have an LG bought in 2019 that does this, and it’s hella annoying when I accidentally hit the input selection button on the remote, switching from HDMI to TV reception.