The heater core is not going to cause over the engine to overheat no matter whats wrong with it.
Engine cooling requires:
- Flowing coolant: If the water pump is failing, the water isn't going to flow through the radiator. If the thermostat isn't opening, the water isn't going to flow through the radiator. If the radiator is plugged up, the water isn't going to flow through the radiator. Given that you've already done the t-stat, I would double check radiator flow. You should also be able to access the inside of the water pump with the t-state removed and see if it looks internally damaged.
- Air over the radiator. The fan handles this when you're stopped and any forward motion 10mph or more will provide enough air to cool it as well. Unlikely this is the problem unless you have a sheet of plastic shoved in your grill.