Spidey has plenty of villains to go around that can make for a compelling story. I think they can get away with not touching Green Goblin or Doc Ock until a third film (although I believe Norman Osbourne should be introduced prior to such). Between The Lizard, Kraven, Electro, Mysterio, Rhino, Shocker, Vulture, Tombstone, Silvermane and Morbius, I think there's enough to tide moviegoers over until having/needing/wanting to replay the big guns that were trotted out already. There are also improvements that can be made with Norman and Otto that would serve to vary things up from the Dafoe and Molina versions, when the decision is made to go in that direction.
A total origin story isn't really necessary. The GA is familiar enough with who/what Spider-Man is to not need it spoonfed to them. Batman '89 and The Incredible Hulk are two movies from this genre that come to mind that managed to convey the hero's beginning without being overly bogged down in detail.
Besides, audiences are going to connect this to the previous 3 films, regardless of what efforts are made to distinguish. Joe Moviegoer isn't following this stuff that closely and doesn't really care. He just wants to be entertained.