The brower, a hypermedia client, sees the links, for example, in the API responses and renders them for the human to interact with. The browser is a hypermedia client, working against a hypermedia API. It not understanding the content of the responses beyond the hypermedia layer is by design: that's the uniform interface of REST.
I mean, this is quibbling over definitions.
I agree entirely with your general point that REST hasn't worked out well as a network architecture for non-hypermedia clients.