As you can see, the character goes down on the rotating platform but not properly up - it will not be pushed by the “pusher” and it will not move along with the platform…
I tried something similar to your game: moving block, rotating platform and a character controller. It worked fine with Is Kinematic and Use Gravity unchecked - gravity makes the platforms fall, and kinematic objects produce no collisions (but can block the character movement, exactly like in your game). You MUST also disable movements in all axes but the one the thing is moving or rotating. It can be done in the Constraints topic: check everything in Freeze Position and Freeze Rotation, except the option related to the movement or rotation of the platform. If you don’t do that, the platforms will react to the collisions and fly away in some direction, like objects lost in the outer space.