Not running the stock bypass hose from the intake manifold to the pump without using a drilled thermostat, will result in the problem you describe. Without either one, coolant doesn't properly circulate through the engine. Depending where you have your temp sensor for your gauge located you will see higher than normal temperature before the thermostat will open.
Having the water valve closed blocks one of the other routes that allows proper coolant circulation in the engine.
I dont know why, but I figured a small block in stock configuration.
So, what engine ?
With the controls set to heat, the valve will open allowing hot coolant to flow thru the heater core. With the heater fan on, the engine might run cooler but it aint enough to properly cool an engine by itself.
In AC mode, (or, I think, other than heat mode) dampers in the heater box should actuate to redirect airflow so it does not pass over the heater core. (A control valve, if present, should remain closed). If the dampers are not operating that might explain it being hot inside the car.
But that all seems a seperate problem from the engine running hot.
The primary coolant system, (radiator, hoses , etc.) should regulate engine temp regardless of flow or no flow thru the heater core.
I suspect you already know this, sorry if it seems so basic.
If you ran a heater hose from where the heater circuit comes out of the engine to where the circuit returns to the engine, that is bypass the heater altogether, what happens ? ( I know the hoses are different sizes but if you hold your tongue right, it'll work)