Have you checked what settings you have for the Ai Servo 1st Image parameters..setting it to Release Priority will get your first frame in a burst off quickly, alternately, having it set for Focus priority will cause it to take longer before the shutter will release and can delay shooting. The same settings can be applied to the rest of the images in the burst, in the menu Ai Servo 2nd Image Priority.
These two settings are under the Pink AF menu tab, and the second menu screen, tab 2. Perhaps set both options to Release and Speed and see if it feels the same.
So this issue is before the shutter fires when you do a half press of the shutter. Doesn't have anything to do with the priority settings.