Personally, I would highly doubt a therapist that would just write you a letter for a body-altering, life-changing prescription on the first visit ... or the second visit ...
I think therapists certainly have a responsibility to their clients to (1) get to the bottom of root causes through accepted psychological diagnosis (2) evaluate you from a non-judgmental perspective, and (3) suggest a treatment plan. All the while they should fully take an interest in assisting you to get (mentally) healthy.
If you're going in expecting the treatment plan from the get-go, and getting it, that's a little alarming to me. It's not like a prescription of Xanax (although don't get me started on that ...) you are seeking to change your life (irrevocably) and the therapist just seems like he's doing his job by trying to evaluate you further and then offer your treatment plan.
The thing about you having to come out to all your family is a little dodgy - unless you're under 18. If you're over 18, you're legal, and that shouldn't be a requirement.
I also wanted to add that just because you're paying a therapist doesn't just "buy" you whatever you want. They're in the medical field and do have to adhere to professional standards. Any medical professional that can be be "bought" is not someone who's very trustworthy or concerned about the client's health if you ask me.
Lastly, what child knows the trans diagnosis? I mean, I didn't find out that existed until I was mid teens as well. But if you had experiences you could relate to the therapist, that would back up the diagnosis. Children do not just go, "I'm trans!" until they're older and they discover that it's an actual "thing" and usually when they do, that's when the light bulb goes on in their minds like, "ohhh that explains it". A really good therapist experienced with gender issues already knows this and doesn't expect the person to be self-aware that they were trans from day 1. But in relating your experiences honestly to the therapist, they can certainly see patterns forming and be able to make an evaluation.