Plane was spotted leveled wings, increasingly pitching down after loud two peaks "thunder"
One problem (of the many problems) with this kind of witness accounts is the speed of sound. If the witness is only one kilometer away, it already takes the sound three seconds to reach him. The "thunder" could well be the normal noise of the engines when go-around power was applied and totally unrelated to whatever happened several seconds later to the aeroplane.
Real bangs of jet engines are usually caused by compressor stalls and are extremely rare in modern turbofan powerplants. Overall, they reduce the thrust because the engine will be out momentarily. And even in the worst case that the relight of the engine will result in a overheating of the combustion chamber or the turbine stages, the failure of these components can only result in a loss of thrust, not in a thrust increase. An "afterburner-effect" due to unburnt fuel religting inside the exhaust pipe is not possible with the geometry of a Citation XLS engine. And even a total "explosion" (or uncontained engine failure) can only reduce the thrust and never increase it.
Is there an engine condition capable to generate asymmetrical thrust in order to cause:
Intense YAW to right and PD enough to put a/c outside envelope?
The only one I can think of would be the engine falling off the aeroplane. This would, especially at low speeds, certainly put the aircraft so far outside it's weight and balance limitations that the resulting pitch and roll moments could not be countered with the flight controls. But this can not be seen in the video.