So on those times, being a healer isn't fun.
And WoW found the solution. Hit your "switch to shadow" button, and start melting faces.
The question isn't about classes, it's about roles.
All Lithose's metaphors of supply lines aside, you can have 6 people soloing the mob together, or a group doing the mob. The difference is that a group should be more than the sum of all six DPS in the group. Meaning you can have content that would absolutely destroy a group of 6 dps trying to solo the mob together, yet falls to a group. That requires having complementary strengths. The trinity is an example of complementary strengths. You have one character that is good at enduring damage, one that can make him better at enduring even more damage (and, in EQ, it was usually the holy quatuor: you had one that reduced incoming damage - we called it the slower).
It is not the only possible set of complementary strengths, but it has one advantage: it's so simple (they even called it "simplistic") that even the average WOW pugger can understand it (mostly), and it is applicable to the immense majority of fights (with a few additions here and there).
The problem with the idea of "flexible tactics" and "adaptable AI" and "varied strategies" is that those things are HARD. For the average gamer, I mean. If you end up with a complete different group dynamic necessary for every 40 opposing class, and most of these dynamics can't be done by your group because you don't have a stunner or a snarer, or whatever, because the dynamic that worked on the mob 1mn ago is useless here... the average MMO player's brain will seize. And after a while, he'll simply quit, because he can't keep his 20 strategies in his mind together. Or, well, the mobs are nerfed until it doesn't really matter which strategy you apply, in which case the complementary strengths is gone, and you're back at GW2-ish "6 people trying to solo the same mob together". Which works, but is not very difficult in the end.
So, either you will end up at that point where it's every kerran for himself, or you will have the same trinity/quatuor dynamics emerge, because there's a simple strategy that applies to the majority of fights, and instead of tank/heal/slower/dps, you will have "main kite/second kite/snarer/dps". And even if the core abilities are tied to class skills rather than weapon skills, you will still end up with requirements that someone has to be the "designated" whatever. And he'd better have ranked his kiting abilities to tier 4 and got the gear, or you can say goodbye to that tier 5 dungeon areas you just excavated.