We use chainmail in our medieval camp for cleaning the dishes. It works like a charm on metal pots, especially when they've been put too close to the fire. I've often wondered about using it on cast iron, only a few things made me hesitate.
The medieval camp is an educational display I participate in, where we move into a park and live in the style of 14th Century, including the cooking. The chain mail we use is very different than this modern version. The pattern the rings are connected with are different for a start, and the rings themselves are flat, hammered metal. Then there's the pots, which are not cast iron, but other metals, although most of the cooking is in pottery. The shape and the texture of the chainmail we use gets under the food and lifts it off with minimal effort. I can see these round wire rings (round, as in the wire to make the rings is round, not hammered flat) would scrub away the food, but probably with a lot more effort than the traditional chainmail it's modelled after.
The other problem with using metal to clean pots is that it scratches the pots. One of my favourite features of my cast iron pots is that the inside is smooth. We seldom use metal utensils in any of our pots, and wash them with nothing harder than a wooden scraper. The chainmail could possibly scratch the pot, making it less smooth, and therefore making it easier for the food to stick to the pot, making cleaning harder, making scrubbing with chainmail more vigorous, making more scratches...
ad infinitum
At home, I seldom cook over open fire, so seldom have problems regulating my heat. When time travelling,
people (including me) often underestimate the fire and scorch the bottom of the pot, for this the chainmail is invaluable. I would hate to cook over fire without the security of having chainmail to help clean my dishes, but on an electric or gas hob, I don't think there would be any advantage to it.