What do you suggest other than wet sponge to clean the tip? I usually try and use a damp sponge, instead of a wet one. This is how I learned like 20-25 years ago as a kid though. I never learned how to professionally solder.
"Brass wool", basically a clump of brass shavings, is quite popular as well. I sometimes use that, sometimes use a damp sponge and I don't find any giant difference between the two. Some people say the brass type are better (more thorough cleaning, less thermal difference, etc.) and some say they are worse (wears the plating off the tips faster)
I actually forgot about the brass you're talking about. I've seen it used, but never used it myself. I only occasionally solder though, just a few times a year either to fix something or to make a simple electronic project kit. I rarely do it correctly, I usually just heat my iron up and solder the item, the only thing I do generally is "tin" what I'm soldering first.
