With the caveat I mentioned earlier about hoping that a different lens may improve matters, I went through many of my old photos with NX studio (which I normally don't use) to look at the AF points and while I have lots and lots of little red boxes on birds' eyes and faces when they're walking around, I couldn't find one when they're flying.
Here's probably the most noteworthy example:
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The photo is uncropped, so that bird is plenty large. It was moving mostly laterally but maybe a bit towards me. The head is mostly tucked in, so the long neck shouldn't be screwing things up too badly. I suppose the contrast could be better between the subject and the background, but it's not awful, especially in the area of the head. This was one of a burst of a good 20 shots or so and none of them were on the eye. Some of them were rejected during culling because the point was actually on the wing.
Here's another from the same sequence, the only other one from that sequence I kept:
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Same bird, same day, but a considerably smaller size in the frame and possibly worse contrast with the background:
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Another, different day:
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Here's one of a tiny duck in the frame, terrible contrast with the background, and, you can't see it well here, but the eye is closed so it is the same color as the rest of the head:
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Tiny, tiny bird, almost no contrast:
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Just two more in flight:
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Also, for whatever it's worth, unlike the OP I have a fair number of shots of airplanes and only one or two where it hit the cockpit: in post cases it focused right under the wing.