Ok i will try to explain this. Gun is not always faster than under center passing and has its own problems when running the "quick game". The west coast offense, an offense predicated on quick short passing concepts and excelling in the 2 minute was originally ran and still best ran from under center. The QB being closer to the line makes the throws shorter and quicker. If you line up 7 yards off the ball and snap it then take a 2 yard drop which is common the Throwing Distance for a 5 yard slant becomes a much longer throw than that same throw from under center. The angle is also different and you run some major risks of having the ball batted or a line backer reacting under neath it. The quick out becomes a flatter trajectory through and is a shorter throw which gets it to the receiver quicker. The key to running this style of quick game is having a tall QB though. Big bodied QB's are rare, especially those with arm strength capable of doing this type of passing attack. Because the west coast offense was often ran from the I formation or singleback formations there was always the threat of run or draw in a way that could catch a defense napping on the run. Because the quick passing game was predicated on getting the ball out in under 2 to 3 seconds lineman would often fire out into a defender to keep their hands down instead of fully pass setting. Again not everyone could have a QB that could run this offense so a few other systems took principles of this offense and found ways to implement them.
The run and shoot short game is basically this. When you run the run and shoot from under center it is designed to mimic run and throw quick option routs to the slot receiver as its base through. The half rolls quick 3 step drops and footwork are designed to allow smaller, more mobile QBs to find throwing lanes between the defense and make up for a deficiency in arm strength by throwing to a receiver that is running uncovered. Hawaii has been doing this from the gun this year but in doing so from the gun has increased the time to throw over a traditional run and shoot due to the travel time of the snap to the QB and the loss of precision on the routes ran by their slots. Our quick passing game is based off the run and shoot, and we have used it in the past. Unfortunately we are limited in what we can run in this scheme by the height of our QB's recently. Both JT and TQM are a bit to short for the straight 2 step fire from under center and in TQM's case his arm really isn't strong enough to hit a 5 yard out to the field side from the gun ( a 40 yard throw as the ball flies).
Other teams looked a the west coast offense and said we can't recruit the type of lineman necessary to run the 5 and 7 yard drops from that offense. Moving the QB back into the gun allowed them to make up for weak OL play and give the QB more time to hit routes. The trade off was the running back was not longer a moving threat, and the line had to rely on more zone blocking, runs take longer to execute which gives defenses more time to diagnose and it is easier to tell if its a pass. Add in the longer throws and the fact that smaller QBs will have to move to find throwing lanes anyway as well as a more upright blocking style. Again the quick game isn't really any faster than a quick 2 or 3 step and fire you just have longer throws at different angles. The extreme of this offense comes in the Air Raid variety which takes elements of the west coast offense ( short passing game, fixed routes and complex route concepts on plays) and does so pretty exclusively by adding in screens and behind the line throws to get receivers in space, because running screens to receivers from under center can lead to laterals and potential fumbles you run from the gun to force everything into being a forward pass. In an air raid these quick screens are fundamental to the 2 minute drill as they don't get receivers down field and you can move 5 yards at a time getting on the ball very quickly.
I too wish we still had the pistol flexbone sets in our offense. But I understand why we don't have them. fundamentally they aren't really quicker than a 3 step roll and the run plays don't hit as fast from them which means you almost always want to be undercenter.