Is there a way to keep this from happening without turning off crash effects or airport vehicle traffic?
GSX is not aware of other traffic. If it were, it would create an entirely brand new class of issues, like traffic deadlocks on parkings, etc.
You should probably choose between the two options: 1) turn down the default ground traffic density slider 2) turn off crash in the sim.
Yes, we might probably prevent this in the hard way, by destroying ground vehicles that get close to your airplane while you are being pushed, but I don't think it will look very good, to have vehicles disappearing automatically around your plane.