The person who notices supplies running low is already doing part of the household work.

Households run on a hidden layer of attention: knowing what is nearly gone, what needs booking, when the bin goes out. That work never appears on a rota because it produces no visible action, and the person carrying it looks like they are simply doing the shopping. Noticing is the tiring part, and it goes on constantly.

Find out who does the noticing in your home and take some of it, not just the tasks that follow from it. Own a category outright, keep track of it yourself, and stop waiting to be told when something is needed.