Imagine this instead (basically same story but slightly tweaked):
Campaign alternates mission by mission between Chief then Locke, treading through the same areas. Wherever Chief goes, he activates something, kills someone, betrays the UNSC, etc. in pursuit of Cortana. Blue team go along with him but become increasingly uncomfortable with his actions. Eventually, they abandon him one by one until he is alone (great chance for some sad/tense moments here between Chief and his pals as they part ways - maybe he could be forced to kill one of them to continue).
Locke arrives just behind Chief to find the devastation that he has left in his wake, including a dead member of Blue Team. Locke learns that Chief is about to activate the guardian programme and vows to stop Chief before inadvertently allows Cortana to ensalve the galaxy.
Locke catches up to Chief but it is revealed that Cortana is actually enslaved by the Didact (or some Forerunner baddie) and has been tricking Chief into activating the Guardians, leaving misery everywhere he goes. Locke forgives Chief and they must work together to undo something that he has set in motion.
Alternatively, for the an alternate ending, Chief decides Cortana is worth it and Locke puts him down for the ultimate “Oh wow I can’t believe they just did that” moment.