The other thing to remember in terms of using soap, brushes, etc is the complexity that a real world situation brings to the event.
In an ideal world, a soft brush should logically not damage a hard metal coin much (though there will of course be some friction damage). But in the real world, one does not apply pure soap and pure water to a perfectly clean brush, and a perfectly clean coin. At the very least there is, presumably, dirt of some kind on the coin, that you are hoping to remove with the soap and brush.
What's in that "dirt"? Nobody knows. Are there fine grains of sand in the dirt? There probably is, if it's "environmental" dirt. If so, then scrubbing at it with a brush will be like scrubbing the coin with sandpaper.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis