Cappytan - I get your point but would suggest your chosen examples need a little tweaking.
Re unicorns, there is nothing that says the fossil record must have a record of them for them to have existed. Given the relative special set of circumstances that result in the fossilisation of an animal, there must be countless species that either did not have a single example fossilised or an example has not yet been found.
I would say that there is nothing in the fossil record that matches either a unicorn or even a species with a developmental forehead horn. Also, unicorns are mentioned along with all sorts of other mythical creatures like dragons, Minotaurs, half horse/half men, mermaids etc - none of which have any type of presences in the fossil record. Therefore, there is zero physical and scientific evidence that unicorns have ever existed and circumstantial evidence they are a creature of myth and legend. Therefore I take the view that there is an overwhelming probability they never existed but am not so closed minded that if physical or other scientific evidence did come to light proving the existence of unicorns then I would be happy to change my mind.
If someone wants to believe in unicorns and even spend their life in some religion based around supposed unicorn pronouncements then that's fine but don't get all offended when asked to explain why there is any merit to their unicorn religion given the overwhelming argument that unicorns were never real. If someone wants me to join their unicorn religion then please don't be miffed when asked to explain in detail the evidence they have for the existence of unicorns and how that superceeds the overwhelming lack of physical or other scientific evidence in the public domain so far.
Regarding your point on no evidence of a global flood 4400 years ago, it is a modus tollens argument that the global flood of the Bible never happened as described but it in itself does not prove the non-existence of a God. What it does do is suggest that the Bible record of such an event is incorrect. When taken along with all the other examples of divine intervention that there is zero physical or other scientific (e.g. geological, archaeological, anthropological) evidence for or where the scientific evidence points to a different conclusion (e.g. 607) then it builds a body of evidence that the Bible is full of errors and therefore any claim of divine inspiration is seriously weakened. It also becomes far harder to accept the even wilder claims of a God directed war on wicked humans, people going off to heaven, an earthly paradise, living forever etc., so on and so on. From this you can conclude that God does not exist.
I think people like Dawkins have covered this subject over and over again. His quote along the lines of being a-theist in the same way same he is a-unicornist or a-fairyist sums it up very well.