My success rate over chat is 0%. I just get told over and over that 'Prime guarantees shipping, not delivery. So your package will be shipped within 2 days [or 1 day for 1-day] to the nearest Amazon distribution center. Usually this matches with the delivery date, but not always. The updated delivery date for your order is when you will get it, so we are following our end.'
Give them a link to the terms of service and say "please direct me to the language in your terms of service to that effect." Follow up with "as a paying prime member, I insist on compensation for Amazons failure to meet their guarantee". This has always worked for me. Some reps have pressed back when I insisted on two instances of compensation for one order with two late packages. I replied "I have no sympathy for a multi-million dollar company that fails to meet its obligations." Have gotten several months of prime and gift cards through chat in this way.
They tried that on me today (I've had a bad run this month - I usually have about an order a day)...
Item bought on Monday, with expected -delivery- Wednesday. Late Monday status said "Package has left seller facility and is in transit to carrier".
It's Friday morning now, no package, and status is still "in transit to carrier".
I argued that "Shipped" means the carrier has it. Not that it may or may not be sitting in a corner of a truck for the last week forgotten or unnoticed.