"It is in fact also experiencing a force along the tangent (i.e. the diretion of instantaneous velocity) of the same circle. But, here is where it gets interesting. It is actually experiencing neither of those. It is in fact experiencing a force that is at an angle slightly between the direction of the tangent to the circle and along the radius."
The only force it experiences is radial. There is no tangental component of force. There is a tangential velocity at all times, but the magnitude of that velocity never changes, only the direction, because no work is done - work is given by the integral of the dot product of force and displacement , which are at all time perpendicular to each other, so the dot product is zero. No work is done. Thus no tangential component of force.
"So, if there is a corresponding force pulling inward on the rock (the "Centripetal Force") then there is also a force pulling outward along the radius (the "Centrifugal Force"). In fact, the Rock is pulling on my hand with such an equal and opposite force. So, my hand (and the string) is experiencing "Centrifugal Force". So, "Centrifugal Force DO EXIST!" "
The rock is exerting a reaction force on your hand. But this on no way implies that there is an outward force on the rock. Action and reaction forces are exerted on DIFFERENT bodies. The force on the rock is unbalanced, That is why the velocity of the rock changes direction. You seem to the think the centrifugal force is exerted on your hand. Getting back to the rubber band in TFA, what does it exert centrifugal force force on? There is no hand. And you are abusing an already misused term. Centrifugal force, if it were real, would be exerted on the rock, not the string or your hand.