The only credible (to me anyway) reason to avoid bronze brushes is that aggressive copper solvents can attack the brush and leave green stains on subsequent patches which can be mistaken for an indication of copper fouling in your bore. 
Personally, I use bronze brushes regularly, although I do try to avoid reversing stroke before the brush clears the muzzle.

shooters can eliminate most cleaning wear and damage by cleaning from the breech, using a bore guide and long-shank jags and brushes that closely fit the bore diameter.  I think you are more likely to damage the barrel's crown with universal cleaning rods that are undersized and allow the back of the jag to “catch” the crown when you withdraw the bare jag (the patch having dropped at the muzzle).