Let’s forget about the delay for now. Unplug the appliance, push the button, and plug the appliance again; if it starts running, all you need is a “smart plug”… if it does not start, things will get more complicated.

Does the doorbell have an ethernet port, and is using PoE? Or did somebody just used an ethernet cable because that is what they had available?

No, those are not thermostatic: the metal part should be a pin, that gets pushed down to control flow; that one is a knob, that must be twisted.