Where had the cloak been? What caused it to get like this? What does this signify? Why is it there? What metaphor could it possibly hold? Does the chief not take care of his cloaks?
Everyone who says the cloak serves no purpose is either making a rationalization for their contempt of the series now, or hasn't thought clearly enough on what it's for. Here is the true purpose: Chief is walking through a desert in a heavy suit of armor that is said to no be inoperable in some ways in terms of shields and useability, so the cloak is there to shield him from the sun to prevent him from getting overheated and practically cooking himself, and to protect the joints of the armor itself since the shields themselves are no longer working properly. That's it, quit yer bitchin.