i fond this fouling shot thing intriguing. I know a few people say they do it at the start of a stalk but doesn't that scare everything off? Also, how far off is your clean barrel accuracy? Surely only an inch or so? I can't tell a cold barrel, clean barrel from a fouled one on mine maybe that's just luck.
It's a valid question. At 100 yds, for most, a difference of between 1 to 1.5 inches might be typical so to 100 yds probably is an irrelevance. Mine used to fire an inch and a half high and to the right a little on first shot. That was, until I changed my cleaning regime and omitted the oiled rag through the barrel, instead using a maths soaked rag for the final clean through. Now, POI is bang on. I see no difference. The oiled rag was an old habit that died hard, but even if dry-patching before use, POI always differed for first round plus I started to notice marks on the case heads and twigged that what was happening was that even slight traces of oil left in the chamber were affecting obituration, allowing the case to slam back into the bolt face. When that started happening (and it took a while), I ceased oiling the barrel completely, and cleaned the chamber with meths. No more problems, no fouling shots needed. I also wonder just how much of this first bullet adrift syndrome is down to the shooter as we sometimes consciously treat the first shot as a fouling shot (say when checking zero) instead of taking deliberate care.
Have to confess I never clean mine. They're all stainless and I never suffer from fouling or loss of accuracy. Most copper solvents 'just aren't' and leaving it in there 20 minutes or 20 hours won't remove any real quantity of copper which you'd only put back the first time you shoot the rifle. I think that excessive cleaning is like over polishing your car. You end up doing more damage than good. My rifles are zeroed when they go into the cabinet and still zeroed when they come out. People talk about cleaning them, then have to 'shoot them back in' before they go stalking. Can't see the point in that. I'd make an exception if I was out in truly poor weather, of course, I'd dry them out properly. But I'd never do much more than run a patch or two through. KG12 is the only copper solvent that I have seen physically dissolve copper. If I had blued steel barrels, I would be saying something quite different. I have an endoscope which I run down my bores occasionally and they look absolutely fine, or again, I'd be advocating cleaning them all day long. I do however, check the chambers for deposits etc that could cause tight rounds. I had a .223 that became so badly caked that I couldn't close the bolt. Lesson learned.
When I do clean mine, I tend to run the rod carefully down the barrel from the muzzle end, then attach the brush and pull through in the direction of the bullet.This avoids flicking muck into the chamber. I've never felt comfortable pushing something against the direction of travel although that's probably just me.
Not cleaning probably isn't right. But I'd say that over cleaning is probably damaging; especially if done badly.

+1 After every use
It might be a giggle if anyone had actually said what you quoted. Maybe you're reading a different thread.
I paid £125 for my bore scope by the way. It even takes stills and video. Have a look at some car equipment sites.
Have to confess I never clean mine. They're all stainless and I never suffer from fouling or loss of accuracy.