Does a rotating control surface change the wing area?
TL; DR: It depends on the specific assumptions of the aerodynamic model, but normally wing area is considered constant.

Wing area normally is defined as fixed, and the CL is given so that any control surface deflection results in a change in lift curve, i.e. CL, only. One could equally attribute the change in total lift following a control surface deflection to an associated change in surface area, and then the resultant new CL to go with the deflection would accordingly be of different magnitude, so as to yield the same total lift.

It is a convenient general convention as far as I know, to consider wing area (also very appropriately dubbed as reference area) as constant, because it essentially eliminates one variable from the model.