Historically, since the founding of New Orleans, the batture had been treated as public lands. It served many useful functions. The batture provided space for a public port. This was especially critical considering that at the time, the city was surrounded by wetlands and the only ingress/egress to the city was via the river. In addition, city residents greatly valued the batture silt and used it to shore up levees and create streets. When the river was low, it was used as a public storage grounds. The batture was also a pleasant place for the public to stroll, as there was a nice river breeze. The primarily French New Orleanians were hostile to the new concepts of private property, which they viewed as imported from the common law as New Orleans transitioned from a French to a U.S. colony.
For more information about present-day Louisianians' relationship to the batture, including people who still live on it at various points along the Mississippi's route, check out Oliver A. Houck's book, Down on the Batture (2010).
Plan of the riverfront of the Faubourg St. Marie, including the disputed batture, by Mansuy-Pellitier, 1807. The plan includes a note asserting that the batture was to be used for the public benefit of the city of New Orleans.
Image provided courtesy of the New Orleans Public Library.