"Public" in terms of land use and beaches refers to "public access". A Hawaii Supreme Court ruling in the 80's ruled that every person must be allowed reasonable access to the shoreline ( legally defined as the land between the vegetation line and water). Hawaiians use the ocean for various rituals and it plays a large role in their belief system.
In order to develop the resort, Ko Olina entered an agreement with the city that guarantees the public will have reasonable access to the lagoons. Ko Olina, however, manages, operates, and maintains the lagoons at their own expense, and that is how they are allowed to ban SUP in the lagoons.
For what it's worth, all beaches in Hawaii have rules. For example, dogs and bonfires are not allowed on some beaches. Public does not mean free-for-all.