They provide what you've described as an alternative to what most consider the normal way of representing resource links. It doesn't make any of the claims you are making as to why one is more valid than others.

So once more: sure we can do things your way, but why is it better? Every other source says link headers are an option. You claim they're superior. Why are they better?

Why does representing links in headers and not bodies make the API easier to understand, navigate, or use? How does it clarify the interactions? What do I gain from it?

So far, all you've said is "relationships are metadata" or something, which is not how most people consider relationships, and even if it we're true, it doesn't explain why my response can't contain metadata, as long as it's marked as such.