As I understand the arguments regarding Jack's contribution to the creation of Spider-Man, it is put forth in Joe Simon's "The Comic-Book Makers" and elsewhere that Simon was working with C.C. Beck on a spider-themed hero, specifically an orphaned youngster who is looked after by a cruel elderly couple. That project fell apart and Simon recycled bits of it into the Tommy Troy/ Fly series he worked on with Kirby.

When asked by Lee for new publishing concepts, Kirby offered him something akin to the original "Silver Spider" idea, mixed with the visuals for another unpublished idea of Kirby's called the Night Stalker or somesuch. This hero was a fully grown adult wall-walker with goggles and a gadget-gun.

Stan had Jack go ahead with the idea but rejected the results, a brawny he-man with a web-gun, as being standard super-hero fare and not at all what he was interested in publishing. He then reassigned the story to Steve Ditko who redid the visuals and possibly elements of the basic concept entirely.

I've never read an account that said that the Spider-Man that finally saw print was Kirby's design or work. What is contended is that Lee and Ditko would not have been working on a wall-walking, web-shooting orphan hero with a spider-theme at all had Kirby not brought them the concept in the first place.