I think that most of the problems that ail any franchise over a long period of time can be blamed on bad management. I'm not sure whether I've seen a single NHL franchise that folded because it was a 'bad hockey market', in spite of the team doing well and the owners doing all they could do to keep it afloat, while there are examples of plenty of teams that relocated or folded that could have easily been successful in other circumstances, or teams that came close to folding because they won the lottery: Min. North Stars, Pittsburgh Penguins, et al.
So, assuming that the new Atlanta ownership was similarly competent to the people who are currently running things in Winnipeg, I don't see any reason to suppose that the franchise couldn't be successful there. That said, though, they would almost certainly not have the turnout of the Jets, for two reasons, first on account that fans get disgruntled after many seasons of mediocrity and need some time for the team to show that they've improved, and secondly that Winnipeg has also been the victim of losing a hockey team, and were very ready to embrace a new one when it came. The post-relocation Jets have continued to have problems cracking into the playoffs, in spite of having some good players, so it seems reasonable to suspect that the Thrashers in the same situation would be little better and probably do little to inspire the fans' confidence. The attendance might have gotten better than it used to in the last two or three years, as the team has become more of a contender, but I don't think the Thrashers would have made as much money as the Jets have.
A rebranding wouldn't probably have been out of the cards. The Dallas Stars did it under less serious circumstances, in spite of having a lot of good memories associated with the logo and such in the town (left over from the 'glory days'). For a team like Atlanta whose greatest accomplishment was getting swept in the first round by the Rangers, I dare say that putting some distance between them and their old identity might have been a good thing... not that it likely would have helped much given the results they've since gotten.
The old division alignment was set up to account for the teams as they were, and it's unlikely they would have changed to an unbalanced setup without at least some geographical reasoning for it. I suspect Detroit's ambitions for making it into the East would have had to wait for a few years. By the time the next round of Western expansion talk came along, though, they would have had more of an opportunity to do it, perhaps setting up something not unlike the unbalanced conferences we have today. Columbus might have had to wait, though, until Vegas or Seattle came along before getting to go East. I find it hard to believe even the NHL would approve of a 17/14 conference setup.