Lead certainly is not going to wear the steel in,but that may not be the point.
Folks who are skilled in knife sharpening know what "the burr" is.When you hold your angle and stone the blade to the point where you have met the edge,the steel will feather to a very thin burr.Its not part of your blade bevel.It folds,breaks,and is generally not desirable.Removing that burr is a step in the process. There are various ways to do it.
Some folks use a wine cork,some hard felt,some newspaper.Some just use a different stone stroke.
The steel of the knife will cut a lot of wine cork,but the cork will break off the burr.
IMO,a lot of what "break in" is about is NOT the bore,but the inevitable little burrs of chambering.
With your 22 barrel,maybe,"If it ain't broke,don't fix it" is a plan.
If you feel compelled to experiment,A 22 should be easy.
I suggest looking to DME mold supply ot Gesswein. Lookat grit charts for a grade. I used #9.I don't recall the micron. I suggest about 900 to 1200 grit diamond mold polishing compound. It comes in a little syringe.
The full dia of a 22 lr bullet is the same as the case.Smear a little diamond on some glass .Use another piece of glass to roll a few bullets through the diamond compound,embedding the diamond in the lead.Try not to charge the brass with grit.Wipe off the excess.
I,myself,would not want the bore quite dry,but not wet,either.Off the top of my head,poking a very loose patch or maybe a nylon brush with just a little Hoppes on it would be good.Not wet,just a drop or two.
You can have your fun with not much chance of doing harm.I'd slow down at about 15 or 20 rounds. You actually might do some good on a Green Mountain.
But I will say again,trying to improve on what the barrel maker sent you is a separate issue from removing the micro-burrs left by the chamber reamer.