It is guaranteed to happen that as I grab the paper and walk to the person's desk, something more important comes up as somebody sees me, and then I have a meeting, and the piece of paper simply isn't important enough to remember next to my other responsibilities.
Employees are paid to do the work they're best at. It's a bad use of resources to have them tracking office supplies, emptying their wastebins, vacuuming their floor area, restocking toilet paper, or alerting the office manager when milk is low.
For a tiny, cash-strapped startup they might have to do all those things, but generally it's not optimal.
Edit: just to be clear and in response to downvotes, this isn't an ego or self-importance thing. It's just the fact that when in conflict, it is genuinely more important for the company for me to address an urgent technical issue, and show up to an important meeting on time, than to finish running across the office to drop off a post-it note about milk. I already never have enough hours in the day to do everything that needs to get done for my job, and everything is a question of prioritization. I'm just trying to give a realistic perspective here, that we separate out job responsibilities for a good reason.