Doesn't a bore snake have a brass or bronze brush in it?The only time I use bore brushes anymore is when I decopperize a barrel. Any other time I just use a Hoppe's bore snake. It cuts my cleaning time in half. I use bronze brushes when given my pick because my bore solvents will dissolve brass.
If that works for you, it's fine, but brass and phosphor bronze are both considerably softer than your barrel's steel and most cleaning rods are aluminum, which is softer yet. My experience has been that patches alone do an ok job of cleaning a gun that has only fired jacketed ammo, but it takes a brush to get rid of lead fouling.Personally, I don't use either. I take a patch soaked in cleaner, fold it, and push it through the barrel using a wooden dowel rod. I know that isn't answering your question, but I don't want to risk scratching my barrel.