I doubt this is the case.
I have no idea where these theories are coming from, the shot of Richard's board in the trailer/s?
Peter's powers aren't something that were passed down from his father, he wasn't experimented on, he wasn't exposed to radiation from going into the spider room, his powers aren't activated by the spider bite, or anything like that.
Peter gets bitten by a radiated spider and gets powers. That's it.
"An irradiated spider bit him when he got too close, and overnight Parker had undergone a metamorphosis, gaining strength, speed and agility."
This would have been worded very differently if the spider
activated his powers instead of
giving him powers.