hotzenplotz,
The aircraft was probably at or slightly above the OGE gross weight, which means that he couldn't takeoff with the engine power he had, at least not with the steep departure the scene required. When he got out of ground effect in the climb, the power to keep his departure going was more than the engine could provide, so the rotor decayed quite a bit (note the drooping sound especially in the interior video). This means the main and tail rotor are reducing rpm rapidly.
When the rpm goes down, the tail rotor produces less thrust, and more pedal is needed. The thrust is proportional to the square of the rpm, so even a 5% loss of rpm will yield a 10% loss of thrust; .95 x .95 = .9025 This means that if he is near tail rotor limits before the rotor rpm droop, he will run out of pedal during the droop. That is when the spin starts.
The scenario is doubly bad. When the rpm starts down, the torque goes up - even though the power is constant, because torque times rpm is power. Thus, reducing the rpm at constant engine power makes the main torque go up, which needs even more anti-torque! This means the tail rotor margin is being hurt by the torque rise, too, so the spin starts all the sooner!
It looks like the pilot reduced collective to recover rpm and stop the spin, and that caused the crash. It MIGHT have been better to let the aircraft spin slowly and hold altitude and accelerate, but that would take some pretty handy piloting (I did that once in a Cobra, it was like doing a ballet while my heart rate was 120+). Once translational speed is gained, the collective can be reduced and the situation recovered.
This is really not LTE because the spin is caused by a gross pilot error where the rpm is pulled down too low, because the aircraft is in a situation with too little power (and too little pilot planning.)