Why does Percy feel ashamed in The Lightning Thief?

In Chapter 13 of The Lightning Thief, Percy is attacked at the top of the St. Louis Archway. Echidna, the mother of monsters, attacks him with a Chimera disguised as a Chihuahua. Percy ends up being bitten on the leg, and his sword falls through a hole in the archway. Echidna then taunts him that if he were a true hero, he would jump down 600 ft to the water below to retrieve his sword.
At the end of the chapter, Percy jumps, and chapter 14 starts with him hurtling towards the water. On the way down, he prays to his father, Poseidon, and his prayers are answered. When he lands the text says,

Why had Poseidon saved me? The more I thought about it, the more ashamed I felt. So I’d gotten lucky a few times before. Against a thing like the Chimera, I had never stood a chance. Those poor people in the Arch were probably toast. I couldn’t protect them. I was no hero. (Chapter 14)

Percy is feeling ashamed, not because his father had to save him, but because he had put on a poor showing against the Chimera. He was feeling low about his abilities as a hero and demi-god, and he doesn’t understand that his powers are partially what protected him. He is the son of Poseidon, and he has magical powers because of his birth. The shame he feels though is compounded by the fact that Percy is barely hanging on. For many of the challenges on the journey so far, he has barely escaped with his life, and that truth is starting to weigh on his mind. He is constantly questioning himself, his abilities, and what he is supposed to do.
Part of his shame could also stem from the fact that he has, for most of his life, believed he is a monumental screw-up. That type of self-image is hard to banish, and the constant worry that Percy will fail sticks with him. It is only made worse by Annabeth’s constant jibes and Grover’s anxiety.