I think you are getting confused regarding where the air actually comes from, yes it uses the heat from the engine to warm the air up, but where exactly do you think the air comes from originally?
Your attachment says nothing of the actual question, its just about the coolant constantly flowing. It says nothing about the temperature heating up at the same constant whether or not the fans are running.
Now with regards to having the fans off when the engine is cold in the winter and the heater warming up quicker, i'm not a car expert but i do know the basics when it comes to physics, so if you have a constant flow of cold air passing through the system then the energy is constantly being transferred from the hot area (heating system) to the cold area (air itself), thus the heating system itself will take longer to warm up than if it wasn't constantly being fed with cold air.
I don't know how it will work in practice (some cars have auxiliary heaters etc) but the theory is correct.
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