A self-Gödelizing machine can still be out-Gödeled
The Gödelizing operator—to be programmable—must be specified by some finite rule. But in that case, the Gödelizing operator is itself formalisable. The resulting system can be shown to contain a formula that's true but can't be proven in the system.
So the Gödelizing procedure still holds against the self Gödelizing machine.
John Lucas (1961).