Do we have any legitimate evidence of Peter being predestined to become Spider-Man? The only thing I see hinting to that would be the "Do you think what happened to you was an accident?" line in the trailer. However, that line could mean many things depending on the context. There is a possibility that what Connors meant was that Peter surviving the spider bite and getting powers was no accident because they've been working on genetically engineering spiders for a long time. We shouldn't jump to conclusions that Peter was predestined to become Spidey in Ang Lee Hulk style yet. Think about how silly that would be as well. "Oh Peter, it was your father's plan all along for you to go to Oscorp one day, accidentally drop some sheets when the right guy walks by, and then follow him to the room with the spiders where you will get your powers (because I can already assume where he is going and that you will be foolish enough to get so close to the spiders)."
My personal guess? Peter survived the spider bite because his dad created the spiders out of his own blood or made his own blood immune to them. And since Peter has his father's blood, it makes sense why he wouldn't die like the other test subjects have. This doesn't necessarily mean that Richard Parker intended for Peter to become Spider-Man.
If they do go the predestined route, I am going to be extremely dissapointed. Spider-Man is not Superman. He was not destined to do greatness since he was a child. Peter gained his powers entirely by accident and eventually came to the conclusion that he should use them responsibly. That is always what I loved about Spider-Man's origin. The idea of Peter being predestined to become Spider-Man is unnecessary and I really hope they don't go there. It was stupid & pointless when JMS did it in the comics and it would be stupid & pointless here too.