The title is the topic.

I currently use a Pro-Shot coated rod but am debating buying one of their stainless ones. For all those who use a Stainless rod because of the softer coated ones picking crap up, wouldn't the patch you are running through the barrel do that anyways? Is there less pressure with the patch that mitigates the effect? Is it more about repeated use since the patch is discarded after one pass? Do particles get so embedded that simply wiping the rods down doesn't work? Is there something I am completely missing?

I don't know where the idea that coated rods pick stuff up came from. The coating is quite hard on my Dewey and it wipes clean like anything else. If someone is worried about pushing crud around their barrel one must ask what crud was in there when they started that they failed to clean out.

I think what you have here is someone spreading FUD and you believed it. Just clean the thing when it needs cleaned and don't let anything but bullets pass through the muzzle.

I can assure you, it is not merely a Fudd rumor. A google search will return tons of debates between long-range or benchrest shooters on the matter.

Patches get replaced, while the rod doesn't.

Whatever slight stuff the rod picks up is on there until it's cleaned.

Clean the rod with some solvent and you're GTG.

Debates on the internet are not definitive indicators of the reality of any situation. Actual experience, of which I have decades, does seem to help though. The cleaning rod is smaller diameter than anything else you put down the barrel. Secondly, anything that MIGHT be rubbed against the bore by the rod has already been rubbed against it by the brush or the patch so all this worrying is for naught. If your rod is all covered with grime from a pass down the bore then stop shooting whatever ammo you're using. If it picks up any hard particulate at all (which is not realistic for a number of reasons) you'll have a hard time pushing on the rod hard enough to cause it to bow and touch the inside of the bore with sufficient force to do any damage at all. Barrels are made of steel, not meringue.

After a pass, I wipe my rod down with a clean rag.

This comes from rimfire benchrest shooters. The theorie is that the small gritt and crap over time get imbedded in the coating and starts wearing the rifling like a lap.  One of the big time smiths swears it wears out barrels. He says a rod made of carbon steel harder than the barrel like an old Parker and hale rod is best because it won't pick up gritt. They also clean their bores more than most any other disiplines, 99% of the shooters clean after every target so they are making way more passes at a match than the average guy cleaning his bore, so there is more of a chance for damage  I see the point but I use a Dewey anyway. I have a barrel that has over 150,000 rounds threw it and it still shoots very good with no damage.  I do replace my coated rods for my match rifles after 3-4 years.

Every pass you make you wipe the rod.  I've got coated rods over 15 years old that look like new.  It's a non issue unless you make it one.