That's not how APIs are used. APIs consume and provide data. Raw data is unsuitable to be presented to the user. That's why HTML has so many formatting options. Formatting information is completely missing from APIs.
> Seems to have worked more or less for 30 years.
Yes, worked for good old web. In this sense, true REST is nothing new and even seems backwards. If we try to do REST while keeping data and presentation separate, we will come to something very similar to XML for data + XSLT for formatting. Or XForms. Old ideas all over again.
> I don't understand that, actually. Markup is underrated for data exchange potential these days.
XML/markup does not map well to basic data types in current programming languages. These work with strings, ints, floats and arrays/dictionaries thereof. Not Nodes, Elements and attributes of unknown/variant data types.