The Suzuki Method in America became the mainstream way of teaching violin and cello since about the 1980's. Before then, there was very little understanding of pedagogy or way of training musicians to teach children. The Suzuki Association of the Americas trains musicians to become better teachers and it provides opportunities to families in educating their children.
I would say that good Suzuki teaching is a way of learning the musical instruments of Western Classical Music in small steps, through the senses and through the imagination, in an environment where the student can develop self-confidence through their ever-increasing ability to express themself. The benefits are many but ultimately, if love is the highest priority in the relationship between parent, child and teacher, beauty will pour forth from the soul of the player into beautiful sound.
My experience with the Suzuki Method
I've had Suzuki teacher training through short-term courses in both the US and in England. I was a "Suzuki parent" of my 3 children for more than a decade so I've observed countless Suzuki lessons with many teachers on cello, violin and piano.