It's ALWAYS the older franchise. Had the Giants/Dodgers stayed, I'd say by now you might see one of those two with a bigger fanbase outside the city, especially when you figure that long dry spell the Yankees had when they essentially had the market to themselves. From 62 to 77 and 78 to 96 the Yankees had no WS titles. And from 65-75 and 82-95 didn't even play in one. Just 8 years after getting the market all to themselves, the vaunted Yankees went through a dry spell. And an even bigger on after back-to-back WS titles. Their mystique was gone for a long period of time.
Mets were brought in to try and split the market up, but mostly just got Long Islanders and Brooklynites that were never fans of the Yankees either. If the team went to Brooklyn, you might have seen a much bigger nostalgia push. But a team in Queens was not competing with the Yankees. Even if the Bronx is a garbage borough.
NFL? Same problem. Giants had decades on the Jets. And had one of the earlier NFL teams survived like the Yanks, maybe they'd have a share of the market. Playing in NJ at "Giants Stadium" certainly didn't help. Jets were the LI team. And then abandoned them for a stadium far from their base fans.
Knicks/Nets? This might be the one that Nets could've taken a crack at the top spot in the market. The Knicks fans have had a rather bitter view of their owner, and despite the prestige of 'The Garden'.... the place is a dump and badly lagging behind every other venue in NBA. Barclays Center and bringing in some big names early seemed like a win-win, and then the team just because a 'good but not great' middle of the road team and that honeymoon phase is over. Plus, a team wearing black/white is a tough sell without any accent color for neutral fans to just wear apparel just because.
Rangers/Islanders/Devils? Same problem with the Knicks and their venue, but they've always been "New York's Team", and by association the biggest fan from the city globally as well. Islanders have and always will be Long Island's team, hence the name choice and the venue far out east originally in Nassau County outside the city. And they had a great run. Devils have always just been the NJ team and moving to Prudential in Newark has given them more fans in NJ, but cost them any bandwagon fans they had had from NYC that would go over in the Brodeur era because they were just the better team and 'in' at the time (even Seinfield used the Devils as the facepaint gag.
Islanders really lost a lot of prestige moving from tiny Nassau Coliseum to Barclays and abandoned most of their original fan base. Much harder to get to, and the basketball only setup that plagued other NHL teams (think Coyotes), they got their new venue right at the city limits at Belmont Park, but it's still never going to be the biggest fan supported team in NYC or the metro area in general. But they're going to just continue to carve their niche out and be content with it.
Soccer? NYCFC immediately. Red Bulls, formerly MetroStars, have always played in NJ, started as the NY/NJ MetroStars before dropping the location, and the move to the Red Bull Arena and adopting the NY location again, didn't help draw fans. Putting a team at Yankee Stadium instantly won the market. But NYRB just refuse to give up hope that they can succeed.
