The word mayday is effective at tuning everyones attention to a “life threatening emergency”, particularly the controller.
“Declaring an emergency” is not standard even for the FAA and has been used by US crews to describe everything from a minor fuel transfer fault to a wing falling off. We probably wont know if the controller was a little more ambivalent towards the 737 situation because a mayday call was not given however I suspect in this and other situations it may have been a factor.
American RT is not great and has bitten them on the ass before. A recent example being the RJ cockpit fire in DEN where the controller did not understand “roll the trucks” and did not respond to the emergency appropriately as a result.
This will not be the first time the FAA again reiterate the need for standard RT procedures.