That sounds like too many people to access something in my book.
I'm new to a stake as an STS and was given a key to a cabinet where they stored all the 'tech equipment' - it also included disciplinary documents, and financial records. I kindly suggested to the stake clerk that we move the equipment someplace else, I didn't need access to all the rest of that stuff. I think I'd seriously look at what people are accessing - For instance, the Bishopric doesn't need to access anything, really, they have clerks that can do that. The exec sec doesn't really need access to documents that you'd store in a filing cabinet. Maybe this is a case where we're not applying good principles of 'access' defined in the handbook and by least-privilege.
One way to help determine this type of access is to model it after LCR... if they can get to it in LCR, or not, do they need access to it in a filing cabinet?
“A long habit of not thinking a thing wrong, gives it a superficial appearance of being right, and raises at first a formidable outcry in defense of custom.”
― Thomas Paine, Common Sense