I think continuity developed simply to get people to buy more comic books. If the story "continued" in the next issue or in a different title, the odds increased that the reader would then buy it. Simple, really. Stan Lee took it to extremes and then the characters began to "remember" what had "happened" last issue. Sadly, those extremes are now the "norm."
That's exactly it, of course. Continuity just refers to the property of being continuous, in fictiion the idea that what you read before is informing what you're reading now. It didn't "develop" and nobody "created" it; it's just a tool that can be applied in a variety of ways to varying extents.
The problem is partly, as Jim points out, the idea that there's no longer a single "story," but rather an unending succession of small chapters of a "big picture" that never ends. It has also turned comics into an Easter egg hunt on the lawn of Purgatory, where entire titles seem to exist just so the writer can make a pun based on a forty year old comic.
But more importantly to my sensibilities, there's also the problem that "continuity" has crossed over into what I'd call "syncretism," the idea that EVERYTHING that has come before must directly affect what's happening right this very second, so in mythological style, we have an endless array of reboots and reimaginings to find the "definitive" version of the character or to excuse ignoring all the millions of little details from previous writers, pencillers, editors, and publishers.
(In the back of my mind, there's a fictional ideal that anybody writing for a character would be required to read all prior appearances. They don't need to refer to them or even remember them, but they do need to have been exposed to what has happened before.)
In the Golden Age, I don't think that many of these issues came up, because the business wasn't mature enough to track recurring readers. What sense does it make to refer to something that happened a year ago when there aren't any back-issue bins and you don't know if any of your current readers had any clue the strip existed a year ago? Once you have letter columns and people asking about other books, you now have a better idea of your audience's range and can be more liberal with mixing things up. Note that the insane leveraging of continuity started really happening as the letter columns were marginalized and then discarded--the creators no longer have a barometer of audience sentiment and get their information from syncophant reviewers.
As to Centaur forming a Golden Age team...it's possible, but unlikely as a long-term thing. Yoc pointed out one case where it looks like a good idea, at Prize. Harvey also had a big team-up that basically fizzled (shunted to the text pages). Fawcett had the Crime Crusaders Club that lasted for...you guessed it, a single story. Lev Gleason launched Daredevil's series with the "Daredevil versus Hitler" set of team-ups, never to be mentioned again. There's also an Uncle Sam story, a kind of fairy tale where many other Quality heroes can be seen in a quick cameo.
I don't know why it failed to catch on at any of those companies, but managed to be big at DC, considering that the team aspect happened at DC essentially by accident.
That's not to say that it wouldn't have happened, but I'm not sure that the market was interested, else the Crime Crushers and America's Stars would be better remembered. It may well be that people were just buying All-Star because it had all the popular second-tier characters, rather than because of the JSA prologue and epilogue. Anybody have any thoughts on that?