AWS doesn't benefit from them resolving the ticket in a timely manner... or at all. AWS benefits from being able to put someone in between the customer and the rest of the company. It's like a layer of insulation, that prevents the customer from actually inconveniencing the business or any of the other people who work there.

TAM doesn't necessarily resolve the ticket nor that's their first priority. Their first priority is to be a contact person or a face of a faceless company. I don't believe they're paid by a resolved ticket (all tickets get resolved anyway), MTTR or something similar. Tickets are for some faceless support personnel. TAMs are supposed to help you, but unless it's something trivial they are a waste of time.