In my experience, it seems like native glass has an edge over adapted glass for BIF work, but it's very slight. It's certainly not why the a1 tends to do better with BIF work. Also, remember that it's notoriously hard to test AF performance - especially with something like birds in flight. You would really have to shoot a long time to come to a determination - if you could at all. (Also, "sharp" is a relative term - I see a ton of stuff that people think is sharp that is anything but. Also, is sharp a sharp eye or just anyplace sharp on the bird? Everyone has different standards, making AF comparisons somewhat difficult in this type of discussion.)
I think the bigger difference is that Sony has a better subject detection system for BIF work. Nikon uses body - face - eye where Sony only uses face & eye. What often happens with Nikon is that when you have the AF area on the bird, it doesn't always see the eye or even the face, so it goes for the body. With subject detection, this means that you can have the AF area smack-dab on the face and the camera will focus on the body instead. OTOH, in that situation, Sony will simply use the AF area as normal or favor close-focus priority (really depends on the AF area) so you usually still get the shot.
You can, of course, turn subject detection off with the Nikon, but that is almost never necessary with the Sony, so the Sony tends to get better AF refinement for BIF work in that scenario.
Sony also seems better at sticking to the face or eye of a BIF target, as well as being better at initial AF locks. In fact, I was out yesterday (I'm in Africa at the moment) and on several occasions I put a Wide AF area right on a bird against a blue sky and the Z9 racked the focus ring in and out (800PF) - each time the bird was directly under the AF area with no background / foreground obstructions. By the time it locked on, it was over. Sony usually gets on the first time in that scenario. Of course, most of the time the Z9 did lock on, so don't read anything into that previous statement that's not there. It's not that the a1 is perfect and the Z9 is awful - there is a lot of common ground there.
In short, both cameras are very capable of BIF work, but Sony does put more keepers on my cards and would be my choice if BIF was my main thing. However, that may not hold forever - look how quickly Nikon is catching up.