I agree about bronze brush. I do 4-5 passes with a regular .22 rifle brush, then it takes 3 or 4 tight patches pulled through and it's clean. I too think it's the choke where lead really matters, but I'm really still not certain that cleaning matters in my FX liners including Superior standard. I do replace my breech O-ring every time I clean, since the brush can be rough on it.
That said, in my case I think variations in hold are the cause of my POI shifts. I finally marked the positions of my bipod feet, rear bag, and chair feet, and sit so that my chest is just touching the edge of the bench. I place my center dot at 12:00 on the edge of my bull, then press my shoulder against the stock til it's centered. My POI wanders a lot less than before.
Warren Page, in The Accurate Rifle, says that when you're shooting a good card for God's sake don't get up from the bench until you're done. I think that says a lot.