Now, apply the same principle to an aircraft with a headwind component of 60kts with 60kts TAS. Ground speed is zero. The aircraft executes a max rate 180-degree turn. The aircraft now has to achieve a groundspeed of 120kts in the time it takes to complete 180 degrees.
So what have you proven? Only that an airplane's momentum and/or energy vector changes when it turns. We knew that already. It takes forces to accomplash those changes, and they are supplied by the flight controls (to re-orient the airplane), the engine (to pull/push the airplane in the new direction), and the airflow over the wings (to lift the airplane in a different direction).
As others have pointed out, this happens with or without gusts or wind shear. As I pointed out earlier, gusts (or wind shear, which is simply a gust of a particular magnitude) affect the airplane because they are sudden. You are correct that the airplane's inertia causes the airspeed to vary significantly during the gust/shear.
However, a coordinated turn is not a sudden maneuver. A 180 degree turn may take anywhere from 30 seconds to a minute or more. In any steady-state wind, the control and power response is sufficient to accomplish the required change in momentum/energy vector in a controlled manner with respect to the air mass, and with no regard for the ground.
Only if the pilot changes his frame of reference at an inappropriate time (i.e., fixates on the ground) or if the air flow around the airplane changes suddenly and uncontrollably (i.e., a gust or wind shear) is there any "problem" in completing the turn. Again, if power and indicated airspeed are not increased in the turn, stall margin is reduced, and the effect of any gust or wind shear is exacerbated.