Solve a problem someone has first. Rarely will that solution be an API. Advice is never universally right or universally wrong, but the advice in this post is almost universally wrong.
Technically, API first is a good thing. I don't know about from a business sense.
It was difficult to do this in the past, because browsers could not support the data that returned from the server without a lot of work. Today, JQuery that can present the data with minimum effort.
Don't know if that will be your need since all depend from the service itself if it is useful or not. But we have to bear in mind that Twitter has 75% of its traffic coming from its APIs, while eBay has 8 billion of worth items pushed through the marketplace via APIs. That's the future, especially when the internet of things will come, at that point your toaster will have an API as well.
To be fair, a lot of Twitter's success was due to their API, but without a decent minimal web presence, nobody would have signed up. Remember, everybody's first Twitter client was the web. If that sucked, or didn't exist, nobody would have bothered to pick up an API-based client.
So maybe it would have actually been cheaper in the long run to have built a perfect API first, rather than a web service followed by an API.