How does the transition off pilot to main curcuits happen..By the slides going up and changing the diameter of the venturi..which changes the air pressure to a lower pressure.
This makes a low pressure area around the air jet which sticks up in the middle of the carb throat, which in turn, causes fuel to be sipped up the hole and into the carb and then into the cylinder. ( picture a full to the rim pop bottle and you suck the pop from the rim with your mouth) Your mouth has created a low pressure around the rim of the bottle and you can sip pop (fuel )up from it. This is the basic same principle that the carb is doing.
If the jet needle is too LOW into the needle jet, then you dont get the right timing for the fuel to be sucked up..nor the right amount.
The pilot jet supplies fuel up to AND thru the transition from pilot to main curcuit. So if the pilot jet is too small for the requirements of the setup, you will get a stutter, stumble, call it what you want during the transition. So you need the jet needle setting AND the correct pilot size to compliment each other to effectively make a smooth and strong transition.
As for why the stock airbox is better. Simply put, it supplies a tad bit of resitance to the air flow ( which has been figured at the engineering level ).
This restriction also add to the low pressure inside the venturi to make the carbs suck the fuel up thru the needle jets more efficiently.
If you take the pods off and just lay a red shop rag over the carb throats and start the bike, youll see that it runs up better and with less stumble than it will without anything there at all. This is why these Suzukis always run better with the stock boxes. I will post in a follow up an article that is very informational and will support what I have just said.
Also note that fuel levels in the bowl will also effect if the mixture is lean or rich. Very HIGH fuel levels will make the carbs run richer because the require a leeser low pressure in the venturi for them to sip fuel from the bowls...just the opposite for too LOW a fuel level.
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