You present a false dilemma, suggesting that you only get to do one kind of test. And then you present an argument that black box unit tests should be more important.

But the fact is that you can do both. And they pick up different kinds of errors. I fully agree that all of the tests of the basic interface should be motivated by the interface specification, not the implementation. That is, they should be black. But corner cases and special cases in the code are a common source of bugs, and that one may only reliably figure out where those cases actually are by staring at the implementation. (One may guess without peeking, but one only knows after looking.) So including white box unit tests to smoke out possible bugs in corner cases catches errors that black box testing may not.

Therefore after you write your black box unit tests, there is value in adding white box tests as well. Given a positive return from doing the work, one should do it.

This is also why white-box measurements such as coverage percentage (see Devel::Cover) have value.

UPDATE: minor punctuation and added "white-box" to the last sentence to clarify my point.