What's the benefit of this over tools that you're likely already using to build an API, such as Postman, insomnia or Milkman?
Postman, insomnia, and Milkman are REST clients.
I haven't tried it, but I think it'd be quite awkward to get a server setup from Postman. I think it would at least involve calls to external programs to start a server, or to run against an already running server.
The more apples-to-apples comparison would be using Postman vs the chai tests. I'd argue that writing tests in JavaScript with something like Chai as described in the post is better than postman, because the code is in 'readable plaintext' rather than JSON.
That's exactly what tools like Postman are used for. How are you doing any testing during the development process if you're not hosting your web API even locally?
>I haven't tried it, but I think it'd be quite awkward to get a server setup from Postman
There's no need to as I mentioned that you're going to be hosting it somewhere during development anyway.
You will probably host 1 version which is infrastructure heavy and not very flexible. Take 10 developers working on the API. They all need to test their changes. Automatically if possible. Hosting external elements like the database and others is a pain eased by Docker.