k-2, I have gone over your Langrangian derivation and I am beginning to have doubts about the constancy of w. I found a lot of algebraic errors, starting with the eqm's, at the very top. There are a lot minuses for pluses and vice-versa and these ripple down in your other derivations. Plus, I don't see lamba3 in your substitution of the eqm for x1 into the eqm for x2. Also, I'm not sure how you seperated the equations, I would like to see the rigorous proof. And did this prove the constancy of w before phi at pi/2 or did this prove it for all angles of the rotator? For pre-phi, it is trivial to prove it is constant, you don't even need Langrangian. The tough proof would be after pi/2. If you could prove the constancy of w for all angles, after correcting your errors, I would appreciate it very much. Once you prove the constancy you can stop there in the analysis. I attempted to prove it myself, but again there are a lot of errors, so I did not proceed.You are making fundamental mistakes.
Centrifugal force is NOT acting on the slider. Slider is not rotating. (ω=0 -> FC=ω²R=0) That means, whatever's pulling it, is not centrifugal force.
The force acting on the slider is the tension in the arm connecting slider to rotator. The tension on the rotator side of the arm is the centripetal force that keeps rotator moving in circles. Rotator is accelerating towards the slider, not away. That means the force acting on it is towards the slider. And centrifugal force would have to push rotator away from slider. So centrifugal force is not acting on rotator either.
If you go into coordinate system attached to the slider, there is centrifugal force acting on the rotator, but you are obviously in an accelerated frame of reference. Motion of slider is not uniform, so you expect Fictitious forces.
The way you check if there is a net external force is you look at acceleration of center of mass, and it's clear from all of the above, and even your own analysis, that acceleration of CM in the x-direction is zero. That means, no net force in x-direction. That means, centrifugal force is not involved.
If you look in the y-direction, there is net acceleration and net force. But I account for it by showing that this force comes from the rail. Again, centrifugal force not involved.