It's not the EoD time that won, it's the time that works for the host.
A Fate Egg should have their hatched role listed last, allowing that one to be taken. I don't see why Endow would Endo not-Fate Egg but then Vulture only gets Fate Egg.
The idea of claiming is to not let them get protected, having Golden Nymph in the pool of Vulture abilities is just too good in this setup.