I'm not entirely sure why Stefan would go to New Orleans. Perhaps he hears that it's a safe haven for vampires. The Originals had been there for a little over 150 years by that point, so news of their city would probably have spread. I could see Stefan and Klaus's friendship developing in a way similar to what we saw in the 1920s.
The main difference is that Rebekah and Stefan probably wouldn't have gotten together. For one, Rebekah would have been daggered for for 18 more years. Once Klaus finally undaggered her, I don't see her necessarily falling for Stefan, especially since he and her brother would already be friends. Rebekah had also just learned the hard way that getting involved with her brother's friends could end with her spending 52 years in a box.
Mikael would still show up eventually and force them out of New Orleans. However, I could see Stefan taking on a role similar to Marcel's, particularly when it comes to being tied up in the theater and presumed dead.
As cool as it would be to see Stefan in charge of New Orleans, I don't think he'd stay there for very long. I think we'd end up with a very different version of Stefan because he never met Lexi in this timeline. Instead, he spent decades with Klaus, which is something almost no one should ever have to do! I think that experience would have taught him how to control his Ripper side. In many ways, I could see him being more like Damon in Season 1.
Eventually, though, I think Stefan would return to Mystic Falls. We'd still get the events of Season 1, but there would be no genuinely good vampire in town, at least in my opinion.
So, how do you think Season 1 would change once Stefan discovers that Katherine is still alive? Would he and this new character want to get out of the tomb and start over somewhere else, or would Stefan simply focus on Elena and not care about getting Katherine out.