Playing devil's advocate here, but why not, exactly? You don't see people who refuse to tank complain that they don't get to experience the WAR, PLD or DRK storyline, or people who don't wish to heal rage at how they're locked out of experiencing the WHM, SCH or AST storyline. And it's not like "That's different, those storylines are class based!" because, well, Binding Coil was raid based. Sure, it's easier to get those storylines than to get Coil's, no argument there, but it's still an example of storylines exclusive to a specific type of gameplay (class based gameplay, in this case) with no easy or all-access mode, just as Coil was (only there the gameplay was aimed at those who wished to challenge themselves).
"But Coil storyline was an important part of the Main Scenario, the job and class storylines are not!" Well, not so much an important part of ARR's Main Scenario as tying up loose ends of 1.x, but fair enough. Even so, one could still not see the Coil story and be perfectly able to follow the MSQ. It's not as if the Binding Coil storyline was mandatory to understanding ARR's overal plot. Same with the job and class storylines. Only those are fine being gated behind something (having to play other classes you may not like), while Coil, for some reason, isn't. That's starting to look like double standards to me.
You could argue that "the job and class storylines are accessible to everyone!" but, well, so was Coil's. There's nothing locking people out of Coil (besides having had to beat the original 3 HM primals), so the only thing keeping them out of it is themselves. I'm not saying they have to do Coil, I'm just saying they did have the option of doing it if they wanted the story. Which, again, I know wasn't always so easy in practice. Getting groups together or finding a static before Coil was added to the DF and echo'd wasn't always easy, I know that from experience. Even so, the option was there nonetheless.
Non-tanks don't get to experience the tank storylines. Non-healers don't get to see the healer storylines. DPSers, same thing. Story exclusivity does exist in this game, even outside of the raids. So why is it bad that raiders get their own piece of story content, again?