Why does there need to be a thermal gradient? It seems like the action driving the actuator is the expansion of the wax due to temperature. You don't need a persistent heat flux to get that, the wax just needs enough heat warm up and change phase.
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.
There needs to be a gradient or the wax won't move due to temperature. The motion is the change in state, as you say, but it won't change state if you don't heat or cool it.
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.