I've known vinyl users to complain that they can never get their records to sound clean, even after repeated cleanings. Visited a friend with this concern, and proceeding to watch him try to clean his LP while it was spinning on his turntable. No surprise, when he dropped the needle, it was snap, crackle and pop city!
Even the most high torque tables will grind to a halt when the necessary pressure is applied to clean a 12" LP. This can cause the belt to slip on belt driven tables, or could damage the bearing or burn out the motor on direct drive tables. Not worth the risk for such a poor cleaning result.
While you don't need the LP to spin to get it properly cleaned, it does help the process greatly. I bought this Lazy Susan to serve the specific purpose of providing a stable, turn-able platform for cleaning my LP's, and in that regard, it works perfectly!
I put the Susan on a folded towel to hold it snugly in place, and then on top of a small stool to bring it up to a working height. I put several stick-on foam sections (cut up weatherstripping), on the surface of the Susan, and then placed an inverted 16" aluminum pizza pan on top. Lastly, I centered a 12" rubber vinyl slip-mat on top of the pan and voilà! - instant LP cleaning station!
This solution works well, mainly because the Susan is smooth and stable in operation, no matter how hard I'm pressing down on it to get the brush deep down in those grooves. Although I know this is not as effective as something like a 'Spin Doctor' for cleaning, but it's much cheaper, and much less fuss to use. It's a quick, effective, and simple way to clean your records, although only one side at a time, but since I only listen to one side at a time, it's not a problem. I recommend this product for my use case, or any other spin-able needs you may have.