I never use a bronze brush in my .22LR bores. Cleaning may take several patches, but it's usually not necessary to scrub much. The only rimfire bores that I've used a brush in are those that are pitted...and .22LRs don't often pit unless they're cleaned and stored them in a moist location, like a damp basement.
The biggest problems with .22LR bores are at the chamber end. If you shoot a LOT of .22 shorts, it's possible, especially on really cheap ones, that the chamber will be eroded between the mouth-length of the shorts and the mouths of LRs. I did that when a kid, using a really cheap Remington single-shot. Eventually, the LRs wouldn't extract because the fired LR cases would expand to fill the eroded chamber.
Apart from the chamber erosion, I've seen many .22LRs that were never cleaned and allowed to get wet. The exteriors get really corroded, but the bores were usually quite good, due to the bullets' wax coating.
If you feel the need to clean a 22LR bore, it should be done from the breech end and preferably with a rod guide made to fit the rifle. If it's a semi-auto, a pull-through cleaning rig is good. One can be made from trimmer string, melting and drawing one end to a point and the other, melting and making a blob slightly smaller than the bore, that will allow a patch to fit the bore well.
If you wish to use a rod from the muzzle, you should buy a rod guide that fits over the muzzle. I wouldn't use one on a target-quality barrel, however.