Earlier quoted context omitted.
Pretty sure steel wool will have the same issue as the steel brush.
From personal experience as a dishwasher, old steel wool tends to lose little pieces as it gets more and more brittle. Pieces would get caught in the rivets where the pan handle attaches. I was yelled at more than once because of that—but how else do you clean a pan?
Steel chainmail is awesome for cast iron, and probably safe for anything steel wool is safe for. For pans with coating (enamel or nonstick) where that's not safe, soap, water, and sponge works (sponges have the same shedding problem as steel wool, but usually higher contrast and harder to miss and fail to rinse out when they do shed.)