Why do gusts change the airspeed during landing, but turning (which changes where the wind is coming from) does not?
Before I get into the question about the turning fighter jet, a few words about airplanes' being extremely sensitive to gusts during final stages of landing. If this was actually true, I'd be dead by now. Seriously. Airplanes are sensitive to gusts during landing, I'll give you that, but I think it's exaggerating beyond reason to say it is extreme.

Now, back to one of the most persistent myths in aviation:

The myth of the downwind turn

The jet in your question does not make a full turn from headwind to downwind, but it does not matter, the physics are the same as in the myth of the downwind turn.

As you stated the plane is moving along with the airmass. So, for the plane, aerodynamically there is no head-, side-, or downwind. Only us on the ground can feel it, as we are locked into to the coordinate system of the earth with our feet.

The airplane is locked into the coordinate system of the airmass, and as long as the airmass itself does not experience acceleration, the airplane flying in it can turn in whichever way, and it will not notice any change in the wind direction.

The direction of the wind is locked into the coordinate system of the earth, so in your example there is no change in wind direction. Not for the jet aerodynamically speaking, and not in the sense of navigation (relative wind changes, but you use the same formulas for the calculations anyway). What does change is the groundspeed of the aircraft, and this is the reason wind (and gusts especially) causes some concerns for landing planes. Takeoff is also affected, but not quite as much as landing.