Why does a turbine helicopter need to spin a tail rotor?
Well, that’s complex. Where should I begin?


The main job of the tail rotor is to use its long lever arm and directed force/thrust (horizontal lift) to counteract the torque of the spinning main rotor. The exhaust gas of a Turboshaft engine does not produce a large amount of thrust for the very large amount of torque the rotor-powertrain combination produces. See the previous topic: Does a turboshaft&#39;s exhaust provide any thrust?.
Energy is finite. In simplest terms, any energy taken out of the engine before the exhaust gasses reach the exhaust nozzle will be energy taken out of the thrust generated. Turbofans generate less thrust with their hot-side core exhaust than Turbojets. Turboprops generate a lot less thrust with their exhaust than Turbofans. And, Turboshafts generate less thrust than that. The most important contribution the exhaust has in a Turboshaft is to dispel heat and waste from combustion.
NOTAR (No Tail Rotor) helicopters already exist that use Coandă Slots and ducted thrust instead of a tail rotor. An internal fan is used instead of the exhaust, though.
The anti-torque thrust a helicopter creates must be variable with the ability to fine tune.
If the engine fails, you still need yaw control until you land.
As complex, damage-prone, maintenance-dependent, and expensive the typical tail rotor system is, if it were possible to use your idea with today’s engine technology, it would have already been done.
I am sure there are other reasons. But, great question. Way to think out of the box.