If he'd had 30 traffic tickets, yes, I'd consider it a serious defect, one demonstrating contempt for the laws that we are all expected to follow. Heck, thirty parking tickets would be an issue. The question is not whether he ultimately pays the fines (whether we're talking about taxes or traffic citations); rather, it's the attitude that "I don't need to bother with details that others do."
A similar pattern in my experience comes to mind. I was a member of an organization which conducted its meetings according to Robert's Rules of Order. It was a rather large group, and a time limit was placed on "speeches" from the floor. Speaking overtime was subject to a fine, as I recall, of $5 a minute. We had one member who enjoyed the sound of his own voice, and who made a great show of peeling off $5 bill after $5 bill as he was notified that he was speaking overtime. Asked why he continued to talk overtime and pay his fines, he would always say "because I can." Now, I suppose you could argue that this guy was in compliance with the organization's rules; after all, he paid his fines on the spot. But people resent that kind of behavior. When this individual ran for president of the organization, he got clobbered.
Edwards is a serial offender, with a clear pattern of a cavalier attitude toward the law. True, he has ultimately paid his taxes, plus interest and penalties. So he may have cleared himself with respect to the law, but he'll still be seen (accurately) as a jerk.