I don't think anything is impossible but it almost seems we would have to code differently for that to be viable.
Out of curiosity I've checked out terminal apps for both Android and IOS (Termux and ISH) and I like them quite a bit but would never use them for anything serious. Try them out and how you feel about using them for real stuff - for me, the enthusiasm dwindled and I felt like waiting until I had access to a desktop.
There's lots of room for improvement but they should focus on the more social aspects of Github like Issues and pull request discussion.
I think bia3's point (maybe I'm wrong) is that it would be less buggy if it used the same underlying logic as github.com, but perhaps with a modified frontend (ie. m.github.com like the old days)
Benefit of using an app notification always would come down to platform specific experiences. But replying to github threads over email has been something that I don't have to think twice about
I don't see how Github is much different in this regard. Besides, maybe you want the notifications while on the clock.