Why is only the carbonate anion used to precipitate Barium?
Actually, the textbook should remove this question. You should discuss this with your teacher. It will be a service to future students. The heavier elements of group II elements form insoluble sulfates, phosphates and carbonates.

Now carbonates display another interesting phenomenon. If you keep bubbling carbon dioxide in into a group II carbonate solution (or better say suspension), the group II ions form water soluble bicarbonates. It was a classical experiment of "old school" chemistry. Bubble carbon dioxide into a clear solution of lime, it would become cloudy, if you continue that, it becomes clear again.

Another reason, which may be valid is that you may wish to re-dissolve precipitated barium carbonate after filtration. That would be easy to do so by adding a small amount of strong acid. However re-dissolving barium sulfate or phosphate is very difficult.