Yes. GSEs are GSEs and in their original form they are both sponsored and regulated by the government within the parameters that their federal charter mandates.
It is not sponsored vs control (or lack of it). They are corporate entities that straddle the government and private sector, more or less, depending on the operations under examination. So an either/or approach has trouble when examining the GSEs. They have more of the flavor of both/and than either/or
I do not see how the analogy of independent contractor vs an employee works in this case. Help me out.