I believe the "Bring the player, not the class" mantra exists for 5/10-man content. In 25-man, they do indeed assume you've got the full assortment of buffs and likely the majority, if not each, of the classes represented. A 5-man Heroic can be done with 5 Pallies. You can do some past-content 10-mans with 10 Druids. You need diversity for 25-man.

You need a Holy Paladin or two for Heroic LK, sure, but you don't need all those other things you mentioned. Two Hunters is not a requirement, and if you don't have Rogues or Ret Paladins for stuns there are lots of other classes that can cover for them. Chilblains works fine but so do Desecration, Crippling Poison, Typhoon, Frost Trap, Piercing Howl... there's a variety of alternatives to each of those "requirements" you list that are equally effective. Heroic LK is actually very forgiving on raid comp with the 30% buff out now, aside from the fact that you need bubbles to deal with infest and strong tank healing. You can fill the rest of the raid with almost any comp and still complete the fight.

Anyway, I don't think this article was ever talking about Heroic LK.

The article wasn't addressing encounters like LK HM but seeing my 10 man group's healers struggle with the obscene damage in the LK HM 10 hammered home how much the fight is tuned around a holy paladin/discipline priest composition. It's not required and there are many instances now of guilds doing it with pretty much any combination you can think of but it highlights just how good holy paladins and discipline priests are in their respective niches. Despite the frustration that arises when the required classes aren't around to fill their roles in a particular encounter, I appreciate that Blizzard still cultivates niches for each class and even tunes some of the harder content around those niches.