Here are some Mods (as in modifications) that can be applied to a phpBB2 board to handle a problem of this type by expediting the matters. It would be up to the powers that be to implement some or all of these mods, in part or total. It is not that hard to do...trust me on this. ;0)
Admin Panel for Junior Admins and Mods
Allows you to define junior admins and you can also allow the junior admins access to any control panel features you allow.
Yellow Card Mod
Also known as the "Card System". This also includes the "Report This Post" link that you were referring to -- this is the blue card in this mod. Another nice feature is that it allows you to track offensive posters with the use of a yellow card, which is only visible to Mods and Admin inside of all of the first time offenders posts from that point on. It keeps a detailed running list of these offenders so all Mods and Admin can quickly and easily see who the offenders are at any given time and make notations.
Moderator Control Panel
This mod will add a Moderator Control Panel for all moderators.
Site Staff Mod
Displays who is Mod (or Super Mod) or Admin (or Junior Admin) at your board. I thought I would throw this one in their so it would be easy to know just who to contact when situations like this arise.
Super Moderator Mod
Allows the Administrator to add a 'Super Moderator' with restricted administrative controls. And this I think will be the one they should really look into, so there can be a way for the Moderators to actually take action when the need arises....no matter what time of day or night it is.
All of these Mods are incorporated into the Minerva version of phpBB2 which is a very reliable source of stable modifications done to the board. Some of these are actually written by Niels, which is probably the better known of the phpBB hackers. Actually he is The hacker when it comes down to it.
Anyhow...I thought I would just throw those out there for you to ponder over. It will give you an idea of what can be done to rectify this type of situation so that you can Moderate the board with a little more ease.