What everyone should realize is that ATC is not just one person. You may be flying along quite happily at FL410 in the high level sector, but below that may be multiple low level sectors - say FL280 - FL350 and FL180-FL270 and 6000ft - 1800ft - then there will be multiple sector boundaries that may not match the high level sector boundaries. So when your emergency hits you will talk to the current high level sector controller who is going to first deconflict you from aircraft below (you should of course have situational awareness on traffic immediately below and around you no?
) Then the controller will need to set up the transfer of your aircraft to the next control sector below his and possibly the one below that this takes a lot of talking especially if they have no hint that there is an emergency - they may not even rapidly answer his line.
This is because controllers may have filtered their displays so they don't see traffic outside the vertical boundaries of their sector. If you do NOT put 7700 on the controller in the sector below the one that you are in cannot see you unless the current controller 'points out' your flight in some way (often electronic) but he's also busy talking to aircraft that you may be about to hit.
It really makes total sense for the pilot not flying to select 7700 as soon as possible - then you don't have to tell ATC anything as that 77 breaks through all filtering and all sector controllers will immediately start clearing your path especially when they see your Mode C winding down. They also know not to start telling you about your uncleared level changes or turns as they realize that you are handling an emergency and that communicate comes last. Often controllers will start clearing other aircraft off the frequency to ensure that they can handle your problems.
The 7700 also flags in all supervisor positions including those in other centers and control facilities allowing the entire ATC provision to ramp up ready to assist in any way possible and other controllers will stop doing hand offs to the affected sector(s) etc etc. All that takes is selection of 7700 so don't hesitate to do it and don't hesitate to call MAYDAY (it does concentrate the controllers' minds) you can always downgrade the emergency to PAN once you are level at 10,000ft and handling seems OK etc.