Let's talk about the heater air flow in your system. You are totally correct, there is hot water flowing to the heater core ALL the time. The manufacturers have been doing that since the 1980's because water control valves were hugely unreliable. Instead, they allow you to control heat via air flow thru the heater core. That red/blue control moves a little door (blend door). When it's set to full red, then all the air inside the heating ventilation system is routed thru the heater core. At full blue, no air is routed through the heater core.
Its pretty obvious, that your temperature control / blend door is stuck on full cool. I'm pretty sure your system uses an electric motor actuator. Older, (and often less expensive cars) use mechanical cables to control the blend door.
Its possible the electric motor is dead, or possible that you've stripped the plastic gears within the door controller.
Question for you. With engine off, power on, fan on low blower speed, when you adjust the temperature from full cool to full heat and back again, do you hear any noises coming from behind the instrument panel. If you hear noise, probably a stripped plastic gear. If no noise, dead blend door electric motor.
Depending on your vehicle, this could be a fairly easy repair (perhaps thru the glove box area) or a way difficult repair, involving detaching and dropping the instrument panel. The actuators are normally priced around $25 USD or so.
You mentioned that your "heater core appears to be working". I'm assuming that you've pinched off the heater core hoses with your fingers when the engine is running and can feel warm coolant running through the hose? If not, its possible there are other factors at play here, related to coolant flow thru the core.