The answer to these question will depend on your priorities and use case, and the choice can easily be between no site or a js rendered site.
"I understand that everything in HTML could also be represented in another format. But if your master data is in HTML, does it always make sense to convert the data to JSON just for the sake of it? Would you build a separate client-side rendering component only for that conversion?"
There two non-circular considerations that favor HTML:
1. Web pages still render via DOM. JSON data has to be tramsformed, HTML does not.
2. HTML has semantic capabilities. JSON does not.
Contrary to that you are positing that everything should always be based on or converted to plain html before serving to a browser. That's the point where we disagree.
20 years ago a webpage was just text, but it has evolved in so much more.
I'd be ok with a data site rendering everything from a set of json files. There is more legitimacy in having the presentation done in static html.
Same would go with sites mixing different information sources (twitter, rss etc). You can do the data fetching server side, but the user might prefer having it done client side for a reason or another (transparency for instance).
These kind of sites woyld still be purely informative and yet having them heavily using js makes sense.
I could think of many more situations were generating html from a different format on the client side is the right way to go. Horses for courses.