For many years Nandina domestica, also called Heavenly Bamboo, has been the mainstay of southern gardens. Its lovely evergreen leaf form and beautiful red berries in winter made it a natural choice for foundation plantings. However, in recent years it has shown an ugly side (along with 'foundation plantings'!!) and that has gardeners debating about its continued use. I'm not familiar with any research on the invasivesness of Nandina, but in Austin, Texas it has become a pest in several parks and preserves. It is being removed manually by volunteers.
In my own garden I had several Nandina bushes and you are right, I never saw birds eating the berries. But new plants did appear, and not just those suckering from the mother plant. The berries fell off in spring and floated to their new locations with a heavy rain. My volunteer plants emerged along a stormwater course on the edge of my yard, all neatly in a row. That was the end of my Nandinas!
Use your judgment when deciding to use it or not. I have a friend who has many Nandina shrubs in her yard and she cuts off the berries (she really does!!). That's one way to deal with it. But if your goal is to have a diverse collection of plants in your habitat, I recommend that you plant native seed-bearing plants that offer food to wildlife.