However, this means overriding a private method!
Surely by definition, if you can override a method it isn't private :-)
That's not to say that private methods are useless, but I would want to see a more pertinent example before I could comment on how you are using it.
Where "private" methods win is when you want to hide your implementation decisions from subclasses. If you call your method as $self->_whatever you leave yourself open to having your code broken if a subclass adds its own _whatever method.
You can also break existing code if you add a _something_else method to a later version of your base class when a subclass already has a method of the same name.
Using lexically scoped coderefs or alternate calling strategies means we can code safe in the knowledge that a subclass is not going to change the intended behaviour of our "private" implementation methods.