I would say that story matters at least a little bit. But the presentation and execution of said story was always performed better by Bungie.
With CE, Bungie really just focused on the cool factor and feelings of “badassitude,” so the detailed minutiae of the story wasn’t as important as it would be with their later installments. But with each installment, they were able to keep the overall story consistent and cohesive while still giving it at least a fair bit of expansion.
And they were able to sprinkle in just enough story elements like drama, tension, suspense, etc. without being too heavy handed, which would have left player feeling like they were watching a generic sci-fi drama film rather than enjoying an interactive, player-experience.
But with 343’s entries of Halo 4/5, it feels like 343 just had to do their own thing and try to make THE HALO GAME. But personally, the stories of Halo 4 (and especially Halo 5) feel like a Michael Bay-esque cheap dramatic knockoff of classic Halo. With their entries of Halo Wars 2 and Infinite, I feel like that as far as stories goes, they feel more like Halo because they call back to those classic feelings that the original trilogy invoked (and even Halo 3 ODST and Reach to an extent).
Let’s take two characters from Halos 4/5 and Halo Wars 2/Infinite and compare them.
Halo 4/5: Captain Thomas Lasky: Basic, Vanilla, Bland. Spartan Agent Jameson Locke: Boring, Uninspired
Halo Wars 2/Infinite: Captain James Cutter: Cool. Spartan 092 Jerome: BADASS. Even Professor Ellen Anders, she’s got some sass, wit, etc.
With the former two characters, 343 thought they were going deep, compelling characters by making them all brooding, somber, and painfully doing a LOT of talking. It just wasn’t interesting. Also, Unlike in Halo 4/5 (each of which had a different villain that was retconned out of existence), Halo Wars 2/Infinite had an at least somewhat central villain in Atriox, whose introductory scene, badass.
I think I’ve rambled on long enough with my opinions. TL;DR, with Halo Wars 2/Infinite, 343’s taken better steps in their Campaign narrative that do a better job in invoking the classic feelings coming from the classic-era Halo titles, something they failed to do when they released Halos 4 and 5 and tried desperately to do their own thing.