I don't buy that David is afraid of water, because he spent most of the book around water. True fear is not something that can be overcome that easily.
My theory is that David is afraid of becoming an Epic. That's his nightmare. So, "being an Epic" is his weakness as an Epic.
When Calamity granted him Epic powers, he didn't reject them. But his weakness kicks in all the time, so it is as if he didn't get any powers.
As far as we know, Epic + weakness = normal. Since David is always weak, he is the same as any normal person. He was able to receive Prof's gifts even though he was an Epic.