I do not shoot Canon so I do not have first hand experience. Talking to Canon shooters I work alongside they say they would not expect current Canon eye AF to work reliably with an eye that occupies well below one percent of the picture area.
There seems to me no doubt that Storks and Herons are an unusual shape for birds – and Steve has commented on the challenges getting Nikon early versions of bird AF to work with long neck birds.
It is well known that generally ML AF when it cannot detect a subject requires detail parallel to the short dimension of the frame.
With this type of bird I resolve the tiny in the frame eye problem by aiming auto focus at the neck – and get a get results
Nikon has suggested the Z6 III can detect smaller subjects than earlier Nikon cameras.
Perhaps someone with both a Z 6 III and perhaps a Z8 can do comparisons - and if there is a clear Z6 III improvement hopefully it will filter down to other bodies via firmware updates.
In the background a subject detection AF system needs information from several separate points on the sensor to enable AF to identify with reasonable accuracy a person, a face, an eye, a bird et cetera.
Canon currently claim to be working on "next generation" subject detection - I expect Nikon are also doing this.
In the meantime I do not worry too much about the perhaps 2-3 % of subjects that confuse subject detection AF - having easily learned when to switch AF methods.
I live in "sheep" country. Sheep have a very low contrast dark eye - and a bright yellow high contrast identification ear tag close by.
Local Canon, Sony and Nikon photographers seem not to loose much sleep because eye AF with any of the 3 systems usually chooses the "easier for AF" sheep ear tag rather than the eye.
We recognise any form of autos focus or metering does not get everything 100% right all the time - and quickly adapt to when some input from the photographer can help get a better result.