It's totally possible to shape a bimetallic strip in a way that makes it bistable, thus clicking between the two states, just like an electrical switch which uses springs inside which are normally not noisy either.
One construction I've seen is with a permanent magnet for hysteresis; the strip jumps to the magnet when it gets close, and then requires extra tension to release. The thermostat in my first apartment worked this way (and it looked to me like the strip was actually carrying the current to operate the gas heater; presumably a NC valve).