The heater core is used as a cooling system by-pass. The purpose of the bypass is to maintain coolant circulation during warm-up, to prevent cold or hot spots in the engine.
Heated coolant passes through the heater hoses and past the back side of the thermostat. Without this flow of heated coolant, the thermostat would be in a “cold spot” and opening of the thermostat would be delayed during engine warm-up. This is when you would see the temperature gauge climb to “Hot”, then back to normal after you stop the car and idle the engine.
After the thermostat opens and the engine temperature stabilizes, there should not have been a problem for the duration of your trip, but you said that the thermostat closes during driving. I have never seen a clogged heater core cause repeated temperature cycling while driving but i guess it is possible.
I presume you mean the heater core is now bypassed; i.e. coolant continues to flow in that path, but just doesn’t go through the heater core. If so, either the heater core has a blockage, or an air bubble was lodged there. Ask your mechanic if they’ve tested for a heater core obstruction. That’s usually a quick test. Often a reverse-flush will clear a blockage. But if a blockage is confirmed, that suggests the cooling system probably isn’t being serviced frequently enough.
Could something have been allowing air to enter the cooling system? If air is trapped in the upper part of the engine, the air will fill the coolant bypass hose and put air in to the water pump. Water does not circulate, and the thermostat does not get hot water going past it, so it doesn’t open. Eventually the engine will start to overheat and there will be steam, and the steam will should open the thermostat … eventually.
As a test, if your heat is not working inside the car, it means that your coolant is not circulating and you probably have air in the system.
If you drill a small hole in the thermostat, it may allow the trapped air to escape in to the radiator while the thermostat is still closed. The water pump will suck water from the bottom of the radiator as the air allows the water level to fall. The water will be pumped up to the top of the engine and then water will start circulating again.
Like putting the thermostat in backward or having the hoses routed incorrectly?