To check a fan clutch...

When cold, it should spin freely. Then, get the engine to operating temperature and rev a good bit off idle, say 2500-3000 RPMs. Cut the engine off, the fan blades should stop with the engine. If it overruns, it are bad.

And, just because the coolant is clean, doesn't mean the innards are clean as well. Drain the system and flush with clear water. Get a good chemical flush from a reputable parts store. Follow its directions and see what comes out in the wash. And figure out how to get that shroud back on there. Should be just redrilling the holes...