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 you where they are talking about a spacial thermal gradient and you mean it in a temporal sense.
EDIT: the way I see it, talking about gradients here is nonsense. The wax motor works by expansion/contraction due to point temperature. No gradient is involved. Gradients are thermostat-level thinking, and even then, in the greenhouse scenario, there is no gradient to talk about, because the whole point is just to vent the heat from inside to outside when inside crosses certain absolute value of T, and the only "gradient" that matters is the assumption that outside side is less than T at all times.
things lietrally change its volume maintaining mass when its temperature changes, due to atoms moving more energetically, like something be 4x4x4 inches at 40F, but 4 1/16 x 4 1/16 x 4 1/16" at 200F
this isn't a Stirling engine with a displacer, just a sealed off piston