The kid who reliably refills the water cooler ends up holding the keys to the building.

Responsibility goes to whoever has already been reliable about something small, because that is the only evidence anyone has to work from. Nobody is really assessing the water cooler; they are noting that a thing which needed doing kept getting done without a conversation. Over time that becomes a reputation, and reputations decide who gets asked next.

Pick one recurring small job nobody has claimed and take it for a month. Do it without mentioning that you are doing it. You will be surprised how quickly people start bringing you the things that matter more.