Well even
before he was considered the flagship character back in the 60's you see that Stan and the other writers back in those days often depicted Spidey as easily being able to hold his own against the other heroes of the day. Spidey walloped the entire roster X-Men a couple of times, and a robot duplicate created by Kang even defeated the Avengers (Cap, Iron Man, Thor, Pym and Wasp) the only reason they weren't destroyed was because the real Spidey happened upon the scene and took down the robot for them. These events didn't happen in
Amazing
mind you, it happened in each teams respective title. So if you were an Avengers fan back in the 60's, you were probably slightly pissed that your heroes couldn't defeat one lone Spider-Man robot,
as decreed by Stan himself.
Even the Fantastic Four marveled at his abilities. "If a teenager can be so blamed strong, how strong'll he be when he gets
older?"
But honestly, in most events (Secret Wars, Infinity Gauntlet, Civil War, etc.) Spidey is rarely one of the major players in the resolution of the conflict (It's usually Cap, one of the science guys like Reed Richards, or some cosmic space "god" like Adam Warlock leading the charge and saving the day.) He gets some page time mostly because he's super popular and can be the "everyman" thrust into a "bigger than life situation" whose just trying to wrap his head around it, proxy for the readers.