I wonder if they have enough rotational inertia to cause gyroscopic effects while they're spinning. I wonder if it could be used to stabilize the ship. Or it causes more issues than it solves you can just stop spinning it, just like you can stop it to stop it acting like a sail.

Without running the numbers, just looking at the mass of the sail vs. the mass of the ship, is say the effect is neglible.

I know of a company that makes these as retractable spinning cylinders

The PDF that @danielvf posted mentions gyroscopic benefits on page 20.

In heavy enough weather it's possible to drop sail entirely and still make way on a bare mast, so nah, they might act less like a sail, but they don't stop entirely.

In heavy weather a sailboat sails under bare poles. However, it might actually be a stabiling force in rough seas. My sailboat is far more stable with the sails up. Side to side rolling is eliminated.