Centrifugal forces are what keeps the disc together as it rotates.
That is wrong in both relevant coordinate systems.
In rotating coordinates, centrifugal forces point outwards and the internal stiffness of the disk has to counter them.
In nonrotating coordinates, the elements of the disk need an inwards force to move in a circle, which is provided by the internal stiffness.
For objects on the disk, friction replaces the role of internal stiffness.
In order for the body to travel in a circle, centripetal force needs to be exerted by the disc onto the body, but this is limited by the friction force between the disc and body, and if the body slides on the disc, it moves in a spiral like pattern with both centripetal and tangential acceleration. If the friction force was zero, the body would move in a straight line (assuming it had some initial velocity).
In an inertial frame of reference, centrifugal reaction force would be the real outwards force that the body exerts on the disc. In a rotating frame of reference, centrifugal force is the fictitious force exerted on the body, and there is still a Newton third law pair of forces, the inwards force exerted by the disc onto the body, and an outwards force exerted by the body onto the disc. In a rotating frame of references where the body does not slide on the disc, it experiences no acceleration with respect to the rotating frame.