Fair question! My speculation here comes from thermistors being relatively common replacement parts. (clothes dryers, ovens) As you mentioned, temperature-invariant is tricky, and I think it's often a non-linear but close-enough for the design. I have the impression a failure mode can be if the temperature-response behavior changes over time.

A bi-metal contact is only good for a single temperature, I think? (can be adjustable like thermostats) (I'm sure many but not all dish-washers probably do only need a single temperature...)