Friction is only a small part, the biggest problem is when you move weights on cross bars a good percent of the weight transfers onto the pivots, also if it is the intention to move the weights on the horizontal where it is easy to move them, then there is a very small window of opportunity even at low speed, this problem will only get worse as the wheel speed is increased.
Then there is the cross bars and weights travel limits, there would be a lot of energy lost to collisions and shock on the pivots.
I do not see how the wheel can react like a ice skater arms unless all of the weights move in at the same time, and that will load all the pivots and would need more energy to pull in 16 weight against the centrifugal forces. If you move one set of weights in, and one set out then the wheel will not feel any benefit.
All of the above is before the effects of gravity are taken into account.
I think most agree that the Conservation of Angular Momentum can only be achieved in very low friction environments, if not Angular Momentum is less than conserved.
So where is all the energy coming from to do all this work? It could not be from a light push.
Edit, + E1's quote,
Well, it is a one shot occurrence if not using gravity that is, orbits work just fine, as does angular momentum when gravity is working on them.There are physics laws that prevent the motion transfer from re-occurring, though. It's a one shot occurrence, the same as a gravity wheel. I'm surprised you don't see that. I guess you will when you try it.
How big is your wheel diameter? How much will it weigh?