I remember an old experience with the Sharp rotor. I placed a sort of rope-drive around (although the Sharp rotor is not quite circular in shape), intended to drive a pulley in free spin under the effect of the wind. No rotation.
But when I manually operated said pulley, the Sharp rotor turned easily. At that moment, I thought that the low torque of the Sharp rotor was insufficient to generate rotation by driving a pulley, even in free spin.
However, I just repeated the experiment with a kite airplane with rotating wings utilizing Magnus effect, whose wing profile is quite close to that of a Savonius rotor. The same result: no rotation with the fan when I put a loop around a rim surrounding a wing, while I could easily rotate the Sharp rotor manually by acting the same pulley through the same rope-drive.
I believe that the tension, which is nonetheless necessary, prevents rotation. As you suggest, the tension should mainly be provided by another separate element, like Kiwee’s kite lifter.