Unglamorous, cheap, and the most frequently forgotten items on this page.
A brush beats a sponge away from a kitchen. A brush dries between uses; a sponge stays wet, and a wet sponge in a warm locker is the most bacterially active object you own. The FSA’s advice to “wash or change dish cloths, tea towels, sponges and oven gloves regularly” is harder to follow when there is no washing machine, so the answer is kit that dries rather than kit that needs laundering.
Scourers earn their place with stainless and cast iron, which is most of what is recommended here. A carbon steel or cast iron pan wants a chain-mail or stiff scourer and no detergent; a stainless pan will take anything you throw at it. The one thing a scourer must not meet is a non-stick coating.
Cloths need to dry properly. Microfibre over cotton for this reason: it dries faster, and a cloth that is still damp at the next meal is not clean however recently it was rinsed. Carry more than one and rotate them, so one is always drying.
Replace all of it more often than feels necessary. It costs very little and it is the part of the system most likely to be quietly making things worse.