Hi gentlemen. I'm Dave, and I'm always wrong too. It's been 33 years since I was not wrong.
("Hi Dave")
For your fridge...
You should hear the "snick-snick-snick-snick whoosh" when the pilot lights and the fridge starts. It will turn off on its own when it hits the configured temp, and then snick-snick-snick-snick whoosh when it cycles back on again.
Fridge thermostat controls the heating cycles be it AC Element or the GAS Flame
Older models (smaller cabinet/compact) used a gas control that would regulator size of flame based on temp dial setting Closer to MAX...flame would stay ON
Barney...I think with the DSI fridges, there is no longer any flame (pilot), until needed, to spark, light, heat, and shut down. The earlier model fridges used a standing pilot.
No, It does not always stay on. DSI is only on while cooling on propane. A pilot light refer, the pilot stays lit. Make and model of refer involved would be beneficial!