maybe the approach of "anything goes" works...?
No it doesn't. We're slightly bigger then the 20 to 40 you mention sire_tim, but not much. A long time ago I managed to get (most) people off the ambush tactics generally used when I first took on the role. On the stairwell I would be sweetness and light, but if it's not logged properly I will simply ignore it (unless it's genuinely something major of course - give and take!)
I would suggest a team of two or three might work with an informal approach to Help Desk support, but teams of any size need efficient management, not a fire fighting approach to problem solving. IT management is all about forward thinking not "it's busted so fix it!"
If I can see trends through analysis of the ticket logs then I can be proactive about resolutions. My job as an IT Manager is to manage IT, not fix problems on a break-fix / headless chicken approach to problem solving because users are too lazy, or simply uneducated to the benefits of logging issues properly.