Suzuki style of piano instruction is based largely on kinesthetic learning, muscle memory and learning by ear as opposed to cognitive understanding of the principles of reading music and understanding music theory. It also assumes that both student and parent be extremely dedicated to the practice regimen as it is rote learning.
I have found that students who come to me from the Suzuki method have kinesthetically memorized pieces. This is very satisfying to them and their parents as they are able to perform pieces to the delight of their family and friends. However, if they are interrupted or fumble they cannot just continue. They must return to the beginning and hope their fingers remember how to play the piece. These students are frustrated by the hard work of learning a new piece by reading the music. They would prefer to just play the old memorized pieces.
The other issue is that since Suzuki students do not primarily learn to read music, their skill basis is not transferable to other instruments. Piano is a great starting point because it visually displays the notes in a very logical manner. It should serve as a jumping off point for other or additional instruments. With Suzuki, if you decide to switch or add instruments, you are starting from square one.