It wasn't just that Norman knew his identity. It was that Peter lacked a strong support network for most of his career. He wasn't on a superhero team and his loved ones didn't know he was Spider-Man or that Norman was the Green Goblin. The public didn't know who the Green Goblin was until he was unmasked by Luke Cage in ''The Pulse'' and that was because of Peter keeping both their identities a secret. If Peter was a public hero on a team, he'd have had people who could protect his more vulnerable friends and family or find ways to hide them until the threat was dealt with.
Peter and Matt are said to protect their identities in order to keep their loved ones safe but they've done a poorer job of doing that than heroes with public identities.