I definitely gave this some thought, I think it's not so straightforward though. There is a bit of an impedance mismatch between the things that a bug tracking system emphasizes and captures and the things a project management system captures. Simply porting bugs over isn't a clean solution and can lead to confusion among non-developers when the bugs are highly technical. Github makes it especially hard because there is no support for dependencies so you can't roll up a bunch of individual technical tasks under one umbrella without lots of extra effort.
To be clear: while I can describe the problem, I honestly don't have any ideas what the most elegant or clean solution is. I have never found a satisfactory solution. I end up keeping lots of stuff in my head and writing summary emails.