If you are developing a public service for a global audience, I honestly believe it’s absolutely imperative to support at least JSON and XML and a really, really good idea to also support HTML. The only scenario in which I believe in picking one and going with it is when you are developing a service purely for internal consumption.
As consultants we tend to be working on projects intended only for internal consumption, but that’s probably true of most devs and consumer products that are just providing a back end for their apps. B2B apps and SaaS offerings are far more likely to need an API as part of the business model, so that others can build their own client apps.