I tried a pushback with the default 737-800 just now, and I can't see the smallest trace of stuttering, and I can't see how GSX could cause this, because no object is being created at that time, and if the pause was caused by loading of the wave sound for the speech, having a delay won't change anything, you'll just have the pause after the delay.
Also, all requested sounds are instanced when GSX creates a vehicle, but GSX doesn't have any control on the *exact* moment they are really loaded, all sounds in GSX are made using the standard OpenAL audio library, which in turn calls your audio card drivers, that might handle the actual memory management. Some cards, like the Soundblaster X-Fi series, have in-built memory to cache sounds and pre-load them, for the precise reason to prevent any possible stuttering that might happen when loading sounds when your system is already taxed by other tasks running the background. But, it's entirely out of our control, if your card has such feature, GSX would use it through OpenAL, provided the OpenAL implementation of the audio drivers would use it.
So, as you can see, audio is almost like a black box to us, and it's not as simple as "adding a delay".
This, assuming your stuttering (which I can't see here, and I don't have an audio card with on-board sound memory, I have a default mainboard audio) WAS caused by having to load a sound to begin with.