CEB 444, I am almost hesitant to reply as there are a more experienced guys on this forum.
But I am a glutton for punishment so here goes :P
I use a cable instead of a rod for cleaning my rifles
From what I understand, you will cause less damage during cleaning by only running the brush/patches the direction that the bullet travels. I also have a Ruger 10/22 and a Ruger Mini-14, the receiver on both do not allow for a rod to be inserted from the receiver end.
The bore brush is made of brass, not as hard as your steel barrel. For the brush to do it's job it will be a bit hard to pull through the barrel. Do not allow the brush to stop mid barrel, one long steady pull.
******MAKE SURE YOU ARE USING THE CORRECT SIZED BARREL BRUSH*****
From there I run a solvent wetted patch through, then dry ones till they come out clean. Repeat the wet dry patches, if the rifle is really dirty.
If the rifle is going to sit up awhile between use, wet a patch in gun oil and make one pass through the barrel.
There is also a thing called a pach worm that is made of plastic (google it) most are homemade out of weed eater line, some can be purchased. As I have the cables already I haven't bothered with it. But it is an inexpensive solution.