While I understand your frustration, and to some extent I agree that having too many moving parts can make a machine unnecessarily complex, sometimes it is necessary. From the standpoint of an aggravated customer, the layers of bureaucracy seem unnecessary but consider that you're talking about a $4-5k purchase. That, by ANY standards is a high dollar purchase in most business-to-individual consumer models. Customer Service reps aren't always the most experienced either, so having those layers of bureaucracy is key to prevent situations where entry-tier positions aren't hemorrhaging money by sending out high-value items all willy-nilly.
If I were you, I'd politely and calmly probe for more information on the delay and attempt to explain the urgency if there is any. CS Reps can be receptive to that because they're people too. Usually.
Just don't be rude about it and try not to play the "I'm a loyal customer and I'm dissatisfied with this service" card; I do not mean that in an offending way but all CS reps hear that every day and it's less a bargaining chip and more a point of frustration. Companies want to retain you and the best way for them to help you is if they understand you, the issue you're experiencing, and how you're willing to meet them half-way.
Overall, I hope it doesn't take too long because waiting is a bummer and there's too many good games you could be playing right now :).