The only big "side story" content that really feel entirely irrelevant to the MSQ are the Shadows of Mhach and the Four Lords (which gets a lead-in during an MSQ quest, but hasn't come back into anything since). Even Hildebrand has a fairly important bit of foreshadowing, at least in the level 50 quests.

Gilgamesh summons a primal of Enkidu just by standing near a pile of crystals and wishing really hard. At first glance it seems like just another non-sequitur joke, but it foreshadows all of the revelations we get in Heavensward and beyond about the true nature of primals. To a lesser degree, Enkidu being so weak for a primal also shows that the quantity of crystals available determines a primal's power, setting up Shiva losing to Ravana because she didn't have enough power handy.