It's a nice idea, perhaps we should try it sometime.
The idea that central banks are independent is a joke. The question is who they work for? In econ-speak, what is their objective function? What are they trying to maximize? I'd rather have them printing money to spend on pet projects for people than boosting the fortunes of bankers whose only competence is getting the central bank to boost their fortunes. Quite a competence, to be honest, but not really how it's "supposed" to work.