If you take this argument to it’s extreme I don’t see why you couldn’t use it to make a case for only black box testing. “You don’t need to test the classes because when you test the system as a whole you have also tested the classes that it’s made from”.

To be clear I’m not saying you should only do black box testing, I’m just saying that your argument could be used to make that case as well.

In the end I think testing the private methods will give you more assurance and future proofing of the methods which is much harder to get by testing them indiretly