The main obstacle I can see is access control. Most projects would like to allow issue contributions from anyone but limit commit access to the code (obviously). This doesn't map well to public hosting services' access models; even if you use separate repos for issues and code, (a) at least GitHub doesn't even have an "allow commits from anyone" option, and (b) even if there were such an option, you don't want to allow anonymous users to edit any comment, which corresponds to minimal traditional 'commit access', let alone force push or delete, which usually comes along with it. But you do want to allow users to edit their own comments, which might be possible with custom logic based on signed commits.
So basically one would want a custom 'pre-receive' or 'update' hook. I imagine a suitable script will eventually be included in the git-dit repo, which should work fine for self-hosting, but public Git hosting services don't allow uploading arbitrary hook scripts. (GitHub has webhooks, but they're asynchronous so can't block commits, and as I said, there's no anyone-can-commit option anyway.) I guess projects which want to be on GitHub could host an intermediate repo somewhere which allows pushes from anyone but filters them, then pushes on to GitHub from a trusted account…
Does my thinking make sense?