Well, it's information collected at the server side. So if they really wanted they could just store the data and give you a code, but that would require more storage on their end.
Well, it obviously only happens when something goes wrong. Seems valuable to keep the complexity and external dependencies of your error handing to a minimum.
Maybe its
+ a placebo since already stored in storage system
+ ability to see an outage maybe dropped in the filters cause it was assumed its not user-facing and so assumed lesser severity
+ assurance incase storage failed due to server-storage network unreachable
Oh no! YouTube needs to store a few kb of text! How will they have enough space?