I heard that BE says that breathing is not important. I always heard that breathing and breath support is the only thing that really matters. What's the story?

Breathing is absolutely important, also in BE. Without proper breath support, some of the BE exercises are unplayable.In a more indirect way, your breathing will improve, too. It is right to say that BE does not overemphasize breathing. Mostly because a lot of breathing theory tends to overcomplicate matters. One of the most important differences with other methods is that the order is exactly the other way around.If the lips don't operate efficiently, you can have the best breath support in the world and still suffer from poor tone and endurance. This is one of my own observations in the process. Also, it is very well possible that if your lips work efficiently and you improve your breath support, you can make a big leap forward. So, breathing exercises can be very useful (if your lips move in the right direction).