There were several factors for me that went in to my choice for a teacher.
I certainly agree that someone should not just pick the first person on the list.
For me, I knew I needed a teacher for what I wanted to achieve. I also knew what time of day I'd be able to take the lessons. I went through the profiles, checking how much it would cost me for the number of lessons per month I wanted to take. I then wanted to ensure I could understand the teacher, which required listening to the profile video. I wanted to ensure the method of learning I wanted to use was acceptable to the teacher. I settled on one teacher based upon criteria I needed and then scheduled a trial lesson. During this trial lesson, I knew I had found the right teacher for me.
Now, after many lessons with the teacher, there is a relationship that goes far beyond what I thought might be. I know what to expect during the lessons, and she knows what to expect from me. I look forward to homework. I do more than she requires, and she knows I'll consistently do more than she asks.
If I had just chosen a random teacher that came up first, I might not be in the same position as I am now. To me, relationships are vital. There are bonds in relationships that go beyond mere words. Thanks to my teacher I am becoming confident in my abilities. I know when I make mistakes that she won't make me feel stupid. I know she will correct me in such a way that I learn from my mistakes.