Dont think the 'Geniuses' at the store have any way of fixing the issue.. I would say that it needs a patch - badly..
My observations are as follows:
* 2012 Air - Lion before the software update - projector didnt work - banded display..
* 2012 Air - Lion - after the software update () - projector works, no issues..
* 2012 Air - Mountain Lion - needs restart to detect the projector - if I restart with the projector connected and switched on, it is able to detect the projector and work - same behaviour for both 10.8 and 10.8.1
* 2012 Air - Mountain Lion - with HDMI out instead of VGA cable - able to connect the display output to the TV *without* a restart being required..
[Luckily I got the Mac *just* before ML was released, and luckily I had a TimeMachine backup, so I've simply gone back to Lion.. ..waiting for Apple to release a fix..]
On a lighter note: A little while back, we were at the Apple store to watch the videos created by the kids at Apple Summer camp.. The machine they were planning to use was also running Mountain Lion.. Ironically, the staff there faced EXACTLY the same problem trying to get the output to the large projector/TV.. The person in charge tried a couple of things, and then decided to setup two new iMacs on the tables and asked the people to crowd around the iMacs.. I walked over to him and asked him to restart with the projector connected.. He said yes, he will try, and then continued with the iMacs!! The entire group waited for 20 minutes for the team at the store to connect these two iMacs and then crowded around the iMacs watching..
I have been an Apple user for over 15 years.. ..and to be frank, this is not the Apple I used to know.