Browsers (and most other HTTP clients) already send the Host: header, and that appears to be a pretty good way to mitigate DNS rebinidng attacks. It does not save you from things like <img src> attacks.
One common use of local HTTP servers is with the express intent of hosting web pages (not just an API endpoint) that's rendered in an embedded web browser widget, or perhaps even in a real web browser, because the web is a pretty good way of showing UIs. A common example is CUPS - if you're on macOS or most desktop Linuxes, http://localhost:631 is likely to bring up the CUPS admin interface. So browsers saying "I'm a browser" is not really the (un)authorization you want - many times you want browsers, or embedded widgets made from browser code, to access them. So if the Host header isn't enough, there isn't really a better answer than proper authentication. (CUPS is pretty hardened now, but one good solution for CUPS would be a command-line utility that generated a session key and printed out a URL like http://localhost:631/?auth=a1b2c3d4.)
