I disagree with this.
I'm a mod myself and I don't feel the need that we should have the ability to do this as around 75% of the requests are required an admin such as:
- I'm stuck - This would require an admin to teleport to the player and see if this person is actually hostaged or in jail which may require noclip incase the you cannot teleport to the player
- Unown door - This again, is an admin only command
- Demote - Again, admin only command
- Random demoted - Cannot set job back, admin only command
Unless moderators had access to those commands, there isn't much need in them to be able to handle requests.
I say you just work on acting as a good moderator, get noticed and get promoted if you're lucky.
EDIT: However; I think that if mods/admins do click F1, the "Call An Admin" menu shouldn't appear and only the handling system because it's annoying having to constantly close that menu each time you hit F1.