That's part of the dichotomy of any "fast-paced" shooter-oriented game with any already defined character, though. While a story is needed to hold the plot together, any huge focus on it inevitably detracts from gameplay.
However, since Master Chief is defined right as Halo:CE opens, that alludes to a different style of narrative, one that relies on that defined main character as a driving force behind the narrative, rather than events that shape the player's path such as Mass Effect, where the player responds to changes, rather than acting to create those changes, per se. This is especially evident in the games focusing on Master Chief as the sole remaining spartan, when he's actually not. Halo games have used him as the focus.
As for why Bungie hasn't, we all know of the backlash by the average gamer when plot elements like the covenant war and playing as the arbiter kicked in. This backlash was created at least in part by the huge focus on Master Chief as the primary character of the halo franchise. This, of course, lead to Halo 3's focus once again on Master Chief, with the Arbiter being reduced to a secondary assist role, as well as re-focus the entire plot on Chief and Cortana, the only other "constant" character from Halo and Halo 2.
I guess what I'm trying to say is that Halo has always been set up as a character driven series, rather than focusing on the backstory itself, and judging by the what happened in response to the different releases, Bungie has adapted to not only cater to what their fans want not only storywise, but like was said earlier, gameplay wise as well.