But I think it would be look pretty radically different architecturally - where would such a 'push request' live? You don't have push rights on the repo; nor do you have your own fork.
It needs to work in a Gitty-way - if not GH would anger far more people than one-line contributors. And Git needs access to a repo to which to push.
I suppose a fairly nice solution might be something like: - clone - edit - push --set-upstream origin gh-pr-<my-patch-name>
Github could then respond by mocking push-rights for the repo, but really creating a new 'hidden' repo; and mirroring commits on that branch to a PR opened on the original repo (to which you don't actually have anything beyond read-access).
When the PR merged they could delete the 'stealth repo'; of course if you wanted to maintain your own fork it could still work the way in which it does today.