Yes, automatic connectors would solve a lot. Some users might insist on a bidirectional solution, though it's not necessarily a requirement. In fact, I personally wouldn't want Gitlab to request permission to modify the github copy. At most, I would be okay with it subscribing to feeds and email-replying (no need to oauth) like any github user can.

Speaking of disabling, if gitlab allowed disabling pull requests, wiki, issues, etc. and act like a pure git repo, it would solve a problem github refuses to address, namely a project's users wanting to use github/gitlab but the maintainer hosting the primary repo somewhere else. It would also play well with the D in DVCS, which github sort of breaks by leading users to centralize for convenience. I mean, one day github is down, another day gitlab, and so on, it's just how our networks operate.