We don't mean that you can select 10 or 25 random people and expect to succeed and thrive. The mantra is not "bring any player." The mantra is not "class is irrelevant." Instead, it means that the class (or spec) involved does not trump all other issues, which we felt was increasingly happening in BC.
We honestly heard from a lot of players (and experienced it ourselves) that a group might shelve a good rogue because bringing the shaman for WF was a better overall improvement for dps, even if the shaman was face-roller. That just sounds broken.
I think what happens is that players hear "Bring the player, not the class," and add to that "Well, I play my class flawlessly and have perfect gear," and conclude that any failure is because of the game mechanics and not their own skill or gear (and luck certainly plays a factor in any RPG too).
The intent is actually that when faced with who to invite to a 5, 10 or 25 player run that you can consider the player involved. You should never have to take a terrible priest over an excellent paladin just because the priest has some uber buff or required spell. If your guild has lots of paladin healers, then go run Naxx with them.
The emphasis on encounter design is more about execution, coordination or good old gear checks rather than "at this stage, a mage must cast Counterspell or the boss will wipe the raid."
When we are balancing numbers, we can assume that your group has +haste, +AP and +crit without you having to make sure that means a shaman, warrior and feral druid is with you.