Main issue with doing the raids during leveling is that there are iLevel requirements for queuing in raid finder that you typically don't meet with quest and dungeon blues. So you'd have to have a specific path to have the iLevel required and a group of people to travel with you to the dungeons to get them. It really requires a guild to want to do that from the beginning.
I kind of wish the raids were required for your to progress past each expansion. There's a lot of storyline stuff in the raids that would help keep the lore a little more consistent But that would be difficult to implement in retail. It's possible that's the direction they are going with classic servers. And you will be able to experience at least vanilla content when those release soon™
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