Cool but bad idea
It is an interesting idea, but I don't think it is practical or even useful. You might as well simply use a mail client. Some of the issues to consider include;
1) No ticket history and internal notes
2) How would you provide locking mechanism to avoid double replies?
3) Limited functionality...no assign, transfer, status..etc
4) Complex code/logic needed to hand staff "replies" with no benefit.
5) Security/Access issues..
My point is the feature is not worth the efforts or even useful.