you can drive the car all day with no sc belt, it will just feel like a dog big time.
the sc should spin freely, it will spin with the belt off too, as air is being sucked into it and makes it spin, its normal. there is no linkage to the sc, its spins as fast as the crank will spin it.
when you do spin it by hand, it should not make any noise.
with the belt off, engine off too. grab the sc pulley and turn it left and then right, like snap it, there should be no play, if you feel a click you need a new coupler inside the snout.
have him check the coolant too, maybe it over heated on you and thats why the engine stalled out? we still dont know why it locked up do we? make sure the pulleys all spin and that the a/c clutch is still good? seen them lock a engine up before.