My difficulty with that is that without the fact that the string has zero relaxed length the result to be proved is false. Therefore one must make use of that info about the string, either directly or indirectly. An indirect usage might be by using that the trajectory is circular, but I am not seeing how that swiftly leads to constant speed.What I said was much stronger than that and much simpler. With a simple change in origin, the equations of motion have complete circular symmetry and no longer contain the gravity constant g.
What is there to discuss? (If I seem vexed it is because I am! This is just a simple truth. If you feel the need to describe the springs in gory detail then OK we can do that but it cannot change the result)