I think part of the misunderstanding about fluting is that when you add them....you remove weight. Period. Edit: Unless it's done by a hammer forging process where no material is actually removed and just the effective diameter is increased in which case it WOULD make it stiffer by way of that increased diameter...but the weight would remain the same.
You cannot remove something from a tube and make it stronger by doing so...which then brings in the comparative modifier of 'equal weight' that must be mentioned when talking barrels and fluting or confusion will be had for some people.
One way to think about it that I have used when trying to explain this particular thing: If fluting the barrel makes it stronger, then the deeper the flutes the stronger it will become? If not...why? Then the kicker: if removing some material from the barrel makes it stronger, then a barrel with all the material removed must be the strongest of all!
Thankfully this often turns on the 'light of understanding' or the people just give up the argument and go find someone who will agree with them.