One of the biggest benefits I see to API-first is that it encourages you to document everything well, and that process usually leads to better organization and greater consistency.

As far as fat vs. thin, I worry that this article could be a bit misleading. It's great if you have an API that responds in 20ms as very few people can achieve that. But it doesn't matter much in a world where mobile network latency can easily be 200-500ms. If you're worried about app performance, I'd focus first and foremost on the number of requests you are making and making sure none of them are blocking. The network is the slow part. I know a lot of companies with blazing fast API's and sluggish apps, because they're optimizing the wrong thing.