On most of the cars I've worked with, temperature control is merely an adjustable blend of unheated air and heated air.
As the engine and interior warm up, I keep the temp on HOT, so there is no "cooler" air bypassing the heater core - I want ALL the heat in the air that I can get. Once the interior is warm, I'll drop the temp from HOT to wherever comfortable is (thereby allowing some outside air to bypass the heater core and mix in).
Other cars control temperature by limiting the amount of hot coolant through the heater core. I would think you'd want ALL the hot coolant going there to speed things up a bit.
An oven element (or even your house furnace for that matter) only has ON. Temperature is controlled by cycling ON with OFF to maintain the desired temperature. Most cars do not work that way.
Mind you, you mentioned that the fan is not on until the engine is warm. I am not an HVAC engineer, but I imagine you'd still need to expel the cooler air in the interior.