2002 Toyota Celica Heat not working properly

I have a 2002 Toyota Celica 1.8L with ~75k miles. I bought it used last May & only started using the heater when it started getting cold in October. I noticed that the heater did not work consistently. It would only work if I was going over 40 mph. It's gotten worse since Oct. Now It only works if I'm going over 70 mph for at least half an hour. It also works if I warm up my car (let it idle) for at least 45 minutes before driving. And even then I have to start moving before it starts blowing hot air.

I've taken it to my local shop & they said the heater core is clogged. I'd have to get it replace which is a $900 because it's about 6.5 hours of labor.

My fiancee's father, who used to be a mechanic, said that it sounds like the thermostat is constantly open so the fluid is constantly being cooled and not warming up to warm up the heater core.

And the thing is, I'm not sure if I trust the guys at my local shop. I've heard others talk about how they've tried to rip them off. Anyone else here got any ideas? Thanks.