It often does not work this way in practice. The private methods may have sufficiently complex and contextual behavior that it is difficult to adequately test the private methods indirectly via the public interface. This isn't that uncommon for properly architected non-trivial classes.

In fact, inducing the private methods to be adequately tested via this indirection can require far more test code than testing directly because of the logic required to create suitably parameterized calls to the private method. It also tacitly exposes internal implementation in the test of the public interface, since this is the only way you can push parameters to the private method interface. In cases like this, you should always test the private method directly because it actually improves the abstraction of your tests away from the implementation details.