Sorry. I cannot accept your explanation. You eliminate the possibility of using centripetal force as being part of your work output equation by framing your equation in that manner. If you "a priori" exclude centripetal force, then obviously it seems that centripetal force doesn't require energy to become operant. The energy may not be utilised to generate forward momentum (forward kinetic energy) along the race track (in the car example), but energy is required to keep the car on a circular track (and the car's tires know that).
Consider a simple example.
Imagine traveling in a NYC subway car that is traveling at 40mph on a straight rail track. Imagine that you are standing in the center aisle and holding onto a vertical post. Then imagine what happens when the subway car goes around a tight bend at the same speed. You will have to hang onto that vertical post for "dear life" to prevent yourself from being catapulted down the length of the subway car. It requires "energy" to remain stationary in balance and that energy is the energy required to offset a centrifugal force acting on your body. I would imagine that the subway car also needs to expend energy to stay in balance on its circular track, and that energy is centripetal energy.
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