Edit: also, the actuation from the wax motor only happens while it is absorbing (or losing) heat, even if it later ends up in a steady state with no heat flux.
That's why it's a motor, like in dishwashers, not a thermostat like in e.g. greenhouse windows or radiators.
It doesn't need to be persistent, but without some way to create a state that is different than ambient, it's not a motor.
Also you seem to have a miscommunication with the person replying to where they are talking about a special thermal gradient and you mean it in a temporal sense.