Should not the exhaust of the turbine supply enough byproduct to deflect and maintain yaw with a simple deflector (i.e. rudder/stabs)?
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'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.
The turbine output is very hot and therefore needs to be shielded in some way if it is to be routed through any part of the helicopter. This shielding adds cost and weight, so it's generally avoided. The turbine itself is situated very near the center of gravity of the helicopter, so the turbine output has very little leverage on the helicopter's rotation in its current location.
To use the turbine output, it needs more leverage, so it would need to be routed to the end of the tail to a similar location as the tail rotor (which is why the tail rotor is located where it is - to add leverage). Doing this would mean creating a thermally shielded duct down the inside tail... which would add cost + weight. You also have the problem that you need to control the thrust from the tail to make the helicopter rotate left and right. That's not easy if you are using the output from the turbine because that's pretty much fixed unless the helicopter is ascending or descending.
There is a version of helicopter that uses blown air rather than a tail rotor - see the https://en.wikipedia.org/wiki/NOTAR. This still has the turbine output in the usual place, but uses a fan build into the tail to blow air sideways rather than have an external a tail rotor. This makes the helicopter much quieter in operation and reduces the risk of injury to persons working around the helicopter whilst it is on the ground with the engine running because there is no tail rotor spinning for them to get injured by.
The turbine engine of a helicopter has very little exhaust thrust. They are turboshafts, designed to extract every bit of useful energy for driving the main rotor shaft. From Wikipedia:
A turboshaft engine is a form of gas turbine that is optimized to produce shaftpower rather than jet thrust. In concept, turboshaft engines are very similar to turbojets, with additional turbine expansion to extract heat energy from the exhaust and convert it into output shaft power.
A helicopter tail needs to apply between 10 and 20% of the main rotor thrust in order to compensate for the torque of the main rotor. Re-designing the turbine engine to leave enough hot exhaust thrust for the tail is always less efficient than re-directing the extracted main shaft thrust, for instance with a fan driven by a gearbox.
The NOTAR engine uses such a fan, but not to provide all of the required tail thrust - the downwash of the main rotor is deflected sideways, creating a large part of the required anti-torque thrust.