Testing at the API level is effectively integration testing for any kind of non-trivial API.
Unit testing has particular value because it avoids m*n problems. But that doesn't necessarily mean that m and n need to be separately addressable entities in your API. But if you unit test at the API level, then you're forced to do that.
For example, consider a web browser widget that's part of a UI framework. A web browser is an enormously complex piece of software with hundreds of individual modules doing various things. But the API at the UI toolkit level is much, much simpler; an URL and a handful of properties and methods relating to history navigation. Trying to unit-test the web browser at the API level would be ridiculous; no less ridiculous than having your testing constraints drive enormous complexity into the developer-facing API.
There are layers of API. The appropriate test for a web browser widget is a UI framework is probably render this page in the web browser widget, render it in an actual web browser, and check that the two look the same.
Inside the web browser code there will be more detailed things that should be tested - internal APIs. But the point is that these are APIs and should be built as such; they will likely need to be used by other people working on the same code. Even if not, you need this level of structure so that you can build comprehensible software. If every class can monkey around in the internals of every other one, you get an unmaintainable mess very quickly.
Unit testing has particular value because it avoids m*n problems. But that doesn't necessarily mean that m and n need to be separately addressable entities in your API. But if you unit test at the API level, then you're forced to do that.
For example, consider a web browser widget that's part of a UI framework. A web browser is an enormously complex piece of software with hundreds of individual modules doing various things. But the API at the UI toolkit level is much, much simpler; an URL and a handful of properties and methods relating to history navigation. Trying to unit-test the web browser at the API level would be ridiculous; no less ridiculous than having your testing constraints drive enormous complexity into the developer-facing API.