Since you're convinced that this is an unqualified win, please put together a project plan for switching from our current system to Github.
Err, no. That's a waste of all our time.
It has been stated pretty clearly in this thread by a number of senior community people that we're not going to use a closed source system.
GitLab OTOH is released under a MIT license, so it is an option. I don't know how it compares to other suggested options, but if YUriy wants to propose it it's at least a viable option.
I haven't used Gitlab extensively, but it has a feature set similar to Github and then some [1]. The OSS project does seem active [2], but it is still relatively new.