Solved.
It did turn out to be the fan clutch. I guess I really don’t know how to check it. It was spinning, but the mech said it was freewheeling and not spinning fast enough.
Thanks everyone.
For future reference, run the engine until it’s warm, stand by the fan with the hood open and get someone to turn the engine off. If the fan continues to spin more than three revolutions chances are the clutch is toast.