A "push request" would just be a git-format-patch on the contributor's side and a git-am on the project maintainer's side. Github wouldn't have to create a separate repository - they'd just have to keep an incoming "mail" spool for each repo and show some UI letting the project maintainer review these patches and approve them or deny them. Maybe there actually would be an email address representing "push requests" to the repo, but more likely I'd imagine github would just offer an upload box on the project page where you could post a patch file.

This would be so much easier than the pull-request dance that I bet it would lead to a lot of simple fixes or improvements being contributed by people who otherwise wouldn't get involved at all. Those have value in themselves - but it would also go a long way toward helping recruit new project members, if people could dip their toes in easily before having to go through all the mumbo-jumbo of a private fork and configuring the upstream and all that.