If any body has actually read the article, making a barrels involves some form of cutting which leaves little burrs. Even with hammer forged barrels which are very smooth on the rifling have the chamber cut with a reamer, which will leave burrs around the throat of the barrel —according to the article. I think some barrels - military in particular may have the chamber hammer forged as part of the barrel forging process, but may well be wrong with this.
The point of firing one shot and cleaning is to allow the next bullet to have full friction and to knock off the burrs.
The article suggests a clean 1 shoot 1 regime for less than a box of cartridges. You can do the setting up scope and getting it zeroed in the same process.
By knocking off the burrs the rifle may well shoot better, but will certainly clean more easily.
And I keep my barrels clean as that prevents any chance of moisture causing mischief.
Being sympathetic to any form of machine, I would always want to run anything new in a little gently. Whether its a new rifle / barrel or a new car, or even just new break pads. Indeed on a car I always let it get up temperature before using full throttle.
Certainly when I get a new barrel I will run it in gently, whilst get the rifle set up. I have nothing to loose, but I have the satisfaction of knowing that I have given the barrel the best opportunity to deliver good service. Can I measure it. Probably not - especially as I am probably not a good shot.
But I treat my guns as valuable tools that should be looked after - they are after all expensive and are bought with hard earned money.
But I know plenty who just treat their tools hard, especially when the do not pick up the bill. These are the bread and butter for gun dealers.