Was Mozart born into a rich family?

By no means do they all. However, there are a number of dynasties where generations have composed. The Strauss family had three generations of composers, JS Bach had 20 children, several of whom composed with varying success, Mozart's father composed - yet Beethoven had no family history!

What IS required, apart from talent at the job, is an early recognition of that talent and a nurturing of it. In that way, many who show talent at music in early childhood will learn the discipline required - and develop a love for music to last a lifetime. Those are the people who will compose.