The heat is controlled by the Heater Control Valve (who would guess),
which is located under the air intake grille adjacent to blower.

It is vacuum controlled through the two temperature control knobs, so
it opens whenever one of the knobs are not in fully ccw position, or if
there is no vacuum present due to a defect in the vacuum system (the
vacuum comes from the manifold and is passed through the two knobs in
series and then to the valve),

Once the valve is open, the temperature is adjusted by mixing hot and
cold air through some flaps mechanically linked from the temperature
control knobs.

The problem could be the control of the valve (the valve itself or the
vacuum path), or it could be the heater core (which you already
checked).