His success in Louisiana was largely based on the runoff election system the state had at the time. With all the Democratic and Republican candidates running against each other in the first round, Edwards was typically able to make sure he took the biggest share of the Democrat vote with a support base centered on Cajun, Creole and African American voters. From there it was usually easy to defeat a Republican opponent given the political dynamics of the time.
In a Presidential Election, Edwards wouldn’t have that benefit. Not in the system we have now anyways. With that having been said, had he decided to keep his House seat and plan a path to the Speakership, he might’ve held more national power and influence. I think that would be as far as he could go though; I don’t see him ever getting POTUS even before taking his “ethical concerns” into question. I suppose if somehow he strong-armed his way into the nomination, and went up against a Republican candidate who was exposed as having committed some kind of truly awful crime or some such, it could happen. Edwards did have the most ridiculous kind of luck, particularly looking at his final re-election as Governor against David Duke. So the perfect storm of events isn’t totally out of the question but it seems improbable.