Yes it is; and I'll say it again that in your images you appear to be too far away for it work reliably on the eye. Maybe buy Steve's book on Secrets to the Nikon AF system. Here are two sections on using DX to get closer and the process of Body>head>eye. He describes how he captures that sequence (although for a stationary bird) and the images are in the book. Still applies to moving birds. Next time you are out try to see if you can get eyes locked on at a particular distance and then try to catch them in flight at a similar distance.
On page 178 it says -
As a side note, one nifty trick with distant subjects where the camera
struggles to identify the face or eye is switching to DX crop mode. This
enlarges the image in the viewfinder - and in my testing the camera
sometimes picks up the face (or eye) just a bit easier. I mean, if you have a
distant subject you’re gonna crop anyway, why not do it in the field and
give subject detection a hand?
Further on page 167 he states:
The Z9 can detect the body, face, and eyes - and does so in that order. It
happens fast enough that it often seems like it’s just going for the eye, but
according to Nikon the sequence is body > head > eye. In fact, I’ve often
watched the system refine its AF position by starting on the body, finding
the face, and finally drawing a box around the eye. If you shoot the camera
long enough, you’ll certainly witness this yourself. I was lucky enough to
capture this as shown in the series of screenshots below.