This idea has been around since the beginning of myth and storytelling. We just perceive it in a slightly different light in modern society.
The things you refer to: animal bites, radiation, toxic chemicals, genetic engineering, etc. are all just variations of a theme--the transmutation of man. For a man to become more than man, some type of outside force has to change him. What that force can be is limited by what the audience is willing to believe.
In ancient times, this varied from things like the will of the gods to magical items, like magic-imbued armor/equipment, to magical potions or elixers or mystical spells/chants.
Some of these transmutative forces, like animal bites and fantastic equipment, are still used in modern superhero mythology. But since we now understand the natural world through science, the rules have to be changed a little. Previously magical forces are described using scientific/pseudo-scientific explanations so that they seem more plausible, e.g.
- We know that spider bites don't cause you to develop superpowers, so maybe the spider was radioactive.
- We know that magical potions don't exist, and humans can't be transformed supernaturally, so maybe the hero ingested or came into contact with a chemical that gave him superhuman abilities chemically/pharmiceutically.
- We know that gods don't really exist, so the hero was granted his powers by a hyper-advanced god-like alien race instead.
- ...or perhaps it wasn't godlike aliens, it was genetic engineering by scientists playing gods that turned him into a supersoldier.
- We know that enchanted armor doesn't exist, so maybe the armor is actually made of nanotechnology/unobtanium/phlebotinum that's responsible for its amazing qualities.
- We know that lightning strikes and full moons don't grant special powers, so maybe it was a solar flare interacting with phlebotinum that caused our hero to become super.
- We know our hero isn't gonna find a scroll containing arcane knowledge or magical spells, so instead he discovers scientific knowledge or blueprints for advanced alien technology.
It's not so much that someone came up with this innovative idea of using advanced science to create superheroes, but rather that we live in a culture that understands the world through science rather than magic. We want a little more out of our myths than just "a wizard did it". We want some kind of naturalistic explanation for our fantastic stories because the supernatural just isn't as compelling or satisfying anymore.
And when you think about it, things like animal bites, solar radiation, space rocks, alien beings, mutations, etc. aren't really "advanced science" or science at all. They're just natual phenomena that, we in modern societies, naturally interpret through a scientific lens. The same things viewed through the eyes of an ancient sumerian would again just be "magic".