> Where in that RFC does it discuss _why the fuck_ I would want to put an API link in the headers instead of the body?
In HTTP, URLs locate a 'resource'. Then you and the server do content negotiation, implicit and/or explicit, to select a 'resource representation'. Think of these as different formats for the same conceptual thing identified by the URL. Some formats like HTML can support hypermedia that can have embedded links. Some, like 'text/plain' or 'image/gif', can't.
Link headers allow links from the current resource to other resources to be communicated even if the chosen representation can't communicate links in its body.
If you accepted "text/html", it could have served HTML that embedded these links within the response body, but you didn't accept "text/html".
It can choose to send links as headers, if it still wishes to communicate links.