> if you tell them it's an emergency, they'll stay on the phone with you until the matter is resolved.
That seems like a terrible idea. How many times has someone entered a "CRITICAL URGENT" support ticket regarding not being able to change their background image, or being blocked from downloading a game or something?
Those support calls are screened- "is this a work-stopping event for some % of your business" is a standard screening question. They are also expensive.