> Most web apps today use APIs that return JSON and are called by JavaScript. Can you use REST for such services
You kind of could, but it's a bad idea. A core tenet of the REST architecture is that it supports a network of independent servers that provide different services (i.e. webpages) and users can connect to any of them with a generic client (i.e. a web browser). If your mission is to build a specialized API for a specialized client app (a JS web app in your example), then using REST just adds complexity for no reason.
For example, you could define a new content-type application/restaurantmenu+json and build a JS client that renders the content-type like a restaurant's homepage. Then you could use your restaurant browser JS client to view any restaurant's menu in a pretty UI... except your own restaurant's server is the only one that delivers application/restaurantmenu+json, so your client is only usable on your own page and you did a whole lot of additional work for no reason.
> does REST require a switch to HTML representation ... How such HTML representation can even use PUT and DELETE verbs
Fielding's REST is really just an abstract idea about how to build networks of services. It doesn't require using HTTP(S) or HTML, but it so happens that the most significant example of REST (the WWW) is built on HTTPS and HTML.
As in the previous example, you could build a REST app that uses HTTP and application/restaurantmenu+json instead of HTML. This representation could direct the client to use PUT and DELETE verbs if you like, even though these aren't a thing in HTML.
Thanks for the insight. This very well matches my experience from the top comment of this thread. I added discovery related functionality to JSON based API in an attempt to follow REST and didn't see any benefits from the extra work and complexity. Understanding that REST is inherently for HTML (or a similar hypertext based generic client) and it doesn't make sense to try to match it with JSON+JS based API is very refreshing. Even the article that sparkled this discussion gives example of JSON based API with discover related functionality added to it.