I think that's a really good idea (to write an essay with the main points). I'm also about to start with a new therapist and she actually sent me a pretty in depth questionnaire which I found very helpful. It asked questions about my childhood and upbringing, about my education, about my relationship with my family over time. And many more questions too. I found it very helpful to have these prompts. One of the questions was also something along the lines of "why are you seeking therapy right now?" or "what do you hope to achieve in therapy?". I thought that was really helpful too.

It's the first time I've ever begun therapy with a questionnaire and I definitely found it to be a good tool. Now I feel that, when we meet next week, the therapist will already have a good sense of my main issues etc and it won't feel so overwhelming trying to explain to her what I'm seeking there. Since it doesn't sound like either of your potential new therapists sent you a questionnaire, I think that writing an essay for them is definitely a really good idea.