I think that the spreading of religion by missionaries represents the active effort to bring organized religion, religious social structure, religious law, etc to new cities. While historically settlers may have been individually religious they were often cut off from the religious institutions and communities of the homeland. They also often formed syncretic religious traditions with the local populations which weren't strictly connected to the main branch of religion. Sure religion institutions could develop naturally without much input from the religious center but it takes time without a concerted effort. I think this is well-reflected in the current structure of the game.
EDIT: That said, I am intrigued by the idea of assigning religions to settlers or maybe other units. Depending on how much influence a certain religion has in your civilization settlers could come out one religion or another by chance which would greatly influence the future religion of the newly settled territory without the influence of missionaries. Many of the settlers in the New World belonged to persecuted religious groups. Jewish and Muslims in Iberia, Protestants in Northern Europe, etc.