Sure, we already got Chocobos, Anima, and Airships, so why a stagecoach? For when you just want to slowly admire a scenic tour with other like-minded adventurers of course!

There would be a stagecoach going to and from each camp, all interconnected. For example, a stagecoach would travel from Camp Black Brush to Ul'dah, another stagecoach would travel from Camp Drybone to Camp Black Brush etc

The speed of the stagecoach would be the same as walking, but it can hold many players at once. The current transportation systems are about speed, so why not add one that is social? The player can't control the stagecoach, so all they're left to do is chat with the other fellow adventurers and possibly form a party!