Blend door functions, why no heat? exa...

First, there are two doors inside the heater. one door changes air flow between heat and A/c, the other changes where air is going (vent, floor, defrost etc). You did not mention if when you moved the door by hand, if you got heat when you did that, or, if the motor you removed had vacuum going to it and if it was functioning when you moved the dash control. Next, you don't need to cut holes in the box to check the core. If the hoses going to and from the core are getting hot, likely you have flow going through it. You can do a temporary visual check by replacing one hose with a clear plastic one and observing flow. (don't leave plastic in place as it will get soft when hot and will eventually fail.). You either have a vacuum problem not allowing blend doors to move, a second door that is jammed, an obstruction, internal or external (leaves, mouse nest etc) of the heater core.
If yours has a heater control valve that is installed on the heater hoses, check that as well (failure default is usually full on which will normally give you heat, but they can stick in the wrong position as well).
Last, I know removing the box is a real pain. Generally though, in the end you will need to remove it if the core needs changing. I'd first attempt to flush the core, independently of the system by removing both hoses at the engine and draining the core, then filling it with flush, waiting a half hour or so and then blowing it out both ways with a garden hose.