Perhaps to clarify in case it wasn’t clear, its not that the true CVaR of the distribution can take any value (although it can, provided the rest of the objective stays minimized), but that ζ and η can take any values, and so your expression for CVaR will not equal what you would get if you solved the problem without CVaR and then computed it based on the solution values.
It just doesn’t make sense to use this formulation of CVaR if it does not appear in the objective.