This question can be answered in a way that avoids the specifics of discussing a particular example. A class that inherits publicly starts off with everything that defines its parent's semantics - its public functions, and also its public state variables, if it has any. If none of these are overridden, it conforms to the Liskov substitution principle, and it is a widely-accepted design principle that overrides of these properties should be done in such a way that preserves substitutability.
With private inheritance, none of this applies, unless the programmer chooses to implement (or re-implement), in the derived class, all the public properties of the parent class, in a way that preserves substitutability. As C++ does not require a privately-derived class to implement versions of its parent's public methods and variables, this is no different (other than minor and formulaic changes in the code) than if the derived class instead contained an instance of the parent class as a private member. In particular, with private inheritance, the derived class is not, in any functional or operational way, a sub-type of the parent's type, and if your language treats derived classes as if they are subtypes, an opportunity for misunderstanding and confusion has been created (though it should be noted that unless your language has a way to enforce the semantic validity of subtypes (which C++ does not), then this is effectively a matter of style.)