Using "github" in your domain name limits you to github (many other repository services exist) and makes you a potential target for trademark claims.

Also, I think you are onto something. Git remotes are mostly passive, where you have to pull to see whether there are any updates. There is no subscription model to get notified of changes. You have to actively pull to find out what is new.

You should try to make this as a standalone service, which works with Git remotes, not just a Git hosting party. Then locally, you can pull the repo/commits and check what has updated (diff) and inform subscribers accordingly.