I do like the idea of starting by writing docs because they are very lightweight and it forces you to think through the design of the API. For example, "the params on that route don't look correct" or "this is really difficult to explain, so I think we just over complicated this end point. Let's make it simpler".
One could make the TDD argument that writing tests first would accomplish the same thing, but I think tests are much heavier than docs because certain routes could have dependencies and other similar gotchas. For example, a route may require authentication so you either need to have a fake auth token in the test DB or you need to have your test first obtain an auth token via the API and then hit the end point you really want to test.
If you start by writing docs in something like Swagger, then it should be straight forward to generate API tests from those docs (given, you will still have some gotchas like in the authentication scenario, but if you are generating the tests, then you solve that issue once in your test generator and then the boiler plate is created for you). For example, if you add a required param to a route, you first update your docs to define this new require param and then re-generate the automated test suite. A generated test suite like that could be used for TDD purposes to develop against.
Can you expand on your idea of generating docs from tests? Your tool sounds very interesting. Is it open source for me to look at? If not, how did you have to format/structure your tests so that you knew what information to include in the docs? How did the generation work?
