In his video Laithwaite places a toy gyroscope on a tower on ice, it is intended that the ice will provide a frictionless surface. It can be seen that precession occurs as before with the gyroscope precessing around the tower. Laithwaite claims this shows that the gyroscope appears to exhibit no centrifugal force.
It can be seen that the tower has legs which dig into the ice meaning the assumption of a frictionless surface is not valid. Therefore Laithwaite's claim that the gyroscope appears to exhibit no centrifugal force is also invalid.
This experiment was repeated using an air table to provide a frictionless surface.
Unlike in Laithwaite's demonstration in this case the center of mass of the assembly is seen to remain above a fixed point whilst everything rotates around it. An animation of this demonstration along with animations of other demonstrations can be seen on the flash page.
The air table provides a flat frictionless surface, therefore there are can be no net lateral force acting upon the assembly.
It can be seen from the diagram that the only forces acting are vertical. Newton's first law tells us that a particle remains at rest so long as there is no unbalanced force acting upon it, when applied to a body this is true of the center of mass. Since the gyroscope and stand have no lateral force acting upon them there is no horizontal unbalanced force, this means that the center of mass of the system must stay above the fixed point that it started out at.
The videos show that the overall center of mass of the gyroscope and stand stay in one place whilst the rest of the assembly precesses around it.