E&C don't belong at the top of this list.
If we're talking WWE tenure only (meaning that I can't make my case for The Dudleys at #1 because we discount their ECW tenure,) then my top three candidates for #1 are: The Road Warriors, The New Age Outlaws, and The Hart Foundation.
E&C were great, but their comedy bits were overrated. They were a good wrestling team, but they still weren't the Bulldogs or the Hart Foundation. They won TLC 1&2, but Jeff Hardy was the real star in all of those matches. To be frank, E&C found their success because they were a good team in a time where there was an opening at the top. Edge was an overrated performer, outside of his WrestleMania 22 match with Foley, and Christian is overvalued by the IWC. He's underrated by the WWE, but he falls somewhere in the middle of our expectations and their expectations, in truth. In summation, they're a top ten team, maybe a top five if we only count their WWE tenure, but they're not #1.
The three teams I mentioned, only counting their WWE tenures, those are the teams that I would vote for.
The Hart Foundation were the frontier heel and face team in a time where tag team wrestling was great. Jim Neidhart worked the cameras, and Bret Hart worked the ring. They were the epitome of why tag team wrestling works. They had the look, they had the skills, and they are probably the greatest tag team ever, strictly speaking of the WWE.
The Road Warriors were a great team, and they were insanely popular in the WWF, especially against Demolition, but they were best when they wrestled for the NWA and the AWA. I think that they're probably the, "Hulk Hogans of tag teams," for better or worse of that, but their matches were usually really bad and their promos were hilariously bad, sometimes. Still, I think they're better all-time than E&C, if we count non-WWE promotional work.
Finally, The New Age Outlaws, again, only if we're counting WWE tenures, deserve top honors, as well. They too were a prime example of why tag team wrestling just works. Road Dogg had charisma out of his ass, but he didn't have the look. Billy Gunn had a great look and was a sorely underrated wrestler, but couldn't get over to save his life. When you combined the two, before they joined DX, they were an awesomely fun tag team. In the history of WWE television, they were a great tag team.
There's my best three cases as to who should be over E&C. The WWE is lobbying and kissing Edge's ass, though they were really good, but still, they just weren't #1.