Look up the doctors name on google, see what results you get.
My doctor wanted me to go through an endocrinologist because she wasn't sure how to prescribe hormones for trans people, so yeah I had to wait.
The doctor, if legit, may not give you hormones for even a few months, to make sure that you know what you're doing.
It's true it's a struggle to receive T, however, where I live, as long as the doctor is willing to do it, you don't need a letter, and some people just have more luck than others.