I think you are confusing the browser with the web page. You probably think that the Javascript code executed by your browser is part of the "client" in the REST architecture - which is simply not what we're talking about. When analyzing the WWW, the REST API interface is the interface between the web browser and the web server, i.e. the interface between, say, Safari and Apache. The web browser accesses a single endpoint on the server with no prior knowledge of what that endpoint represents, downloads a file from the server, analyzes the Content-Type, and can show the user what the server intends to show based on that Content-Type. The fact that one of these content types is a language for running server-controlled code doesn't influence this one bit.
The only thing that would have made the web not conform to HATEOAS were if browsers had to have code that's specific to, say, google.com, or maybe to Apache servers. The only example of anything like this on the modern web is the special log in integrations that Microsoft and Google added for their own web properties - that is indeed a break of the HATEOAS paradigm.
I'm not confusing it. I was heavily motivated by business goals to find a general solution for HATEOAS-ifying API definitions. And yes, a web page, implemented in HTML/CSS/JS is a facsimile for it in a certain sense, but it's not self-contained RESTful API definition.
Again, you're talking about a particular web page, when I'm talking about the entire World Wide Web. The API of the WWW is indeed a RESTful API, driven entirely by hyperlinks. You can consider the WWW as a single service in this sense, where there is a single, and your browser is a client of that service. The API of this service is described in the HTTP RFCs and the WHATWG living standard for HTML, and the ECMAScript standard.
Say I as a user want to read the latest news stories of the day in the NYT. I tell my browser to access the NYT website root address, and then it contacts the server and discovers all necessary information for achieving this task on its own. It may choose to present this information as a graphical web page, or as a stream of sound, all without knowing anything about the NYT web site a priori.
The only thing that would have made the web not conform to HATEOAS were if browsers had to have code that's specific to, say, google.com, or maybe to Apache servers. The only example of anything like this on the modern web is the special log in integrations that Microsoft and Google added for their own web properties - that is indeed a break of the HATEOAS paradigm.