Peter was just an ordinary kid who got bit by a spider. The fact that his father created the spiders with his own DNA has nothing to do with Peter. This is not a Hulk situation where Richard Parker was experimenting on his son. Richard Parker didn't force Peter to walk into the room full of spiders, the element of random chance in Spidey's origin is still there. The only difference is that if someone else had been bitten by that spider it would have just made them sick. Peter Parker wasn't destined to be Spider-Man, it just so happens that his father designed the spiders which gave him his powers. Not once do they ever bring up the idea that Peter was destined to be Spider-Man in this film.