Some doctors may still want a letter from a therapist, but not all do.
There are several reasons I can think of as to why a doc would want a letter.
1. They are not convinced you are committed to transition.
2. They have not worked with many trans patients.
3. They are morally opposed to transition.
4. They legitimately fear transition will do you more harm than good.
5. They want/need a second opinion.
I do think more mature patients generally have an easier time explaining with conviction to a doctor they are trans. Remember, mature and old are not synonymous. It means psychologically or physically developed. You can be young and mature.
Katie often talks about going in to a doc in full female mode, and others swear by this method. I just went in looking like a guy... but there was no doubt or question in my voice. I needed HRT. The people who have the easiest time getting fast tracked seem to be the people who are convinced, do not speak in uncertain terms, and frankly, can talk openly to a doctor about being trans like it is the most normal thing in the world. It takes inner strength to transition and many docs may be more willing to follow your lead if you can quickly display such strength.
If a doctor practices informed consent, that is a good sign they have worked with a good number of trans patients. If they are unwilling to start you out or continue your treatment, consider their advice, ask for their reasoning, and consider a second opinion.
Latoya, you live in a region where you should have several options if you ever hit an obstacle.
And remember, your doctor works for you. As long as you are not asking them to harm you, you have every right to make suggestions and even insist on certain forms of treatment. You pay them, not the other way around.
As for your shrink, she may just want your money, she may have reason to think you should keep going to therapy, or she may simply be fascinated by you. It may be a combination of all three. Not every therapist gets to work with someone through their entire transition, and IME, that can make folks like us quite the catch for them. Shrinks are people too. They get tired of hearing the same stories every day from cis folk. We often spice up their day.