...unless we take a more compositional approach and define our application to consist of many "parts" each of which has a public API and an implementation. Well, in that case I would argue that each function can be considered as such a part, having a public API (the definition of its inputs and outputs) and an implementation that can be changed while keeping the public API stable.

Over the time I came reached a conclusion: whenever I have to run some code to be sure it does what it should, I write a test instead. And every often that is a unit test. If I'm highly confident that the code will do what I expect, then I rely on very high level system tests, unless I fear that someone might break the code later by accident without the compiler (or other existing tests) being able to detect that.