One of the materials left in a barrel after firing is glass. Yep, ground glass, put into primers to create the friction needed for firing. That glass can become embedded in soft metal or plastic cleaning rods and act like a file on the bore.
Now it takes a while, and probably isn't any more than a theoretical issue for most of us. But it is a good reason to never push a patch through the barrel, then pull it back through.
In cleaning from the muzzle, it is best to push the rod through, put a clean patch on the tip, then pull the patch through, at least the first pass. A muzzle protector doesn't hurt, either.
As for the steel rod, steel rods are softer than barrel steel, so are actually less of a problem than a brass or aluminum rod with dirt, crud and glass embedded in it.
Also, please avoid bore snakes. They sound good, but routinely break off in the barrel, then people try to drive them out, and get other stuff stuck, aaaaagh!
Jim