I don't think this is true at all. The reason that JSON won is stated quite clearly in the interview: you do not need a software stack to handle JSON. This feature sells itself for any software development team who needs to have frontends running in a browser consuming data from a backend. You do not need an evangelist to convince people to use a feature that is ubiquitous, free, extensively tested, and more performance than anything in the market.
Particularly in the early web, the need for async "liveness" of pages didn't exist, so pages didn't consume backends directly. Everything was handled on refresh or navigation and server rendered as HTML. It was only after AJAX-style web apps took off (post-Gmail, really), that JSON started to get real traction. There were no JSON parsers in C#, Java, Python, etc initially. XML was far more common and SOAP was generally used for cross-endpoint communication.