OK, I am not a fan of utterly clean barrels.
I have spent a long time shooting and posting (elsewhere, a highly respectable Tactical Rifle site, with upwards of 30,000 posts over 15 years), mainly about the physics involved and personal experience resulting from tests employing scientific method. Perhaps some of you will recognize my actual name, Greg Langelius.
The shooters there have evolved a policy to clean only when accuracy drops off. In general, it appears to be a successful strategy. I think there should be a bit more to this, but I also like the aspect where refouling is required but a little, as less rounds overall means longer bore life overall.
I personally refrain from ever firing a bullet down a clean, dry barrel. I liken it to why mechanics use assembly lube when replacing engine rings. I have tried pretreating the bore with Marvel Mystery oil, Lock Ease, Hoppe's Oil with graphite added, Alcohol with graphite added, and I continue to try out other pretreatments. I will probably run out of days before I run out of pretreatments. The graphite preparations appear to reestablish steady-state fouling quickest.
I favor graphite because it has been a powder kernel coating agent in smokeless powders since back in the 1800's. It's there to help tame static electrical charge buildup, but there's also no denying that it a very effective dry bore lube. It is likely the key component in what we refer to as 'carbon fouling'. It is for this reason that I refrain from using moly; there is already a dry lube in use. This lube is already being laid down from the first shot after cleaning, and could well be the key factor in establishing the steady-state fouling condition that renders best POI consistency. Pretreatment only hastens that condition's establishment. IMHO, adding moly only complicates a less well known relationship. Treating bullets should be unnecessary with graphite bore pretreatment.
There is more to this, but this post covers my main core of thinking. I believe it is also a fair explanation, along with serious load development, of why my rifles, nearly all of which are factory stock, shoot competitively. Perhaps a fluke, my factory barreled Savage 10FP .260 shot a 190's F Open score in the final 1000yd round of the 2002 Spirit of America Match.
When I clean, I prefer Gunslick Foaming Bore cleaner because bore scoping has shown me that this stuff really does get it all out after 2 or 3 soaks. I stop short of that many soaks because it think a touch of copper may actually be beneficial.
I patch it out, and don't brush the bore. IMHO, bore cleaning should be a chemical process, not a mechanical one. If I am using a more conventional liquid bore cleaner, I use a nylon brush to assist application, to get the solvent all the way into the grooves.
Greg