A reason not mentioned is that testing public methods only is more black box testing. It's important to do this not just to allow significant changes to the implementation without affecting the test, but also because if you tie testing too much to your implementation you may miss holes in your interpretation of the requirements/interface that are better caught in a black box test.
If you want coverage then private method testing is giving you fake or inferior coverage. Those hard to reach paths are only truly tested via the public interface (i.e. called by the function that will call them in practice, not your contrived test function).
If you want coverage then private method testing is giving you fake or inferior coverage. Those hard to reach paths are only truly tested via the public interface (i.e. called by the function that will call them in practice, not your contrived test function).