Keven - Some, but not all, fish eggs are adhesive/sticky. An adhesive fish egg stuck to a bird's food does not just ""fall off"". Plus if the egg is initially adhesive how did it get pulled off the underwater thing it was originally attached or glued to, and then get newly stuck to the birds foot?. We have had carp eggs stuck to the back of our boat bottom and transom. We literally had to use a putty knife to scrape the off hand rubbing would not remove them.
Fish eggs are adhesive or sticky only for a short period of time in relation to the water hardening process, then the surface part that does not adhere to something looses it adhesiveness. Otherwise everything that was in the water in the vicinity of the eggs during the hatching process/time would adhere to the egg's surface thus making oxygen transfer to the embryo almost impossible. Perpetually sticky fish eggs would be littered with debris as they progress toward hatching and this is not the case. Your "sticky egg - bird foot" theory is full of holes.