Hey Jean,
I just wanted to say that it's very normal for a therapist to look at everything in our life, not just the gender issue. The goal of therapy is to make progress in our lives. Part of that would be to consider improving all areas of our life. For example, when I was unemployed, my therapist really harped on that. The reason was that she saw how much it effected my self esteem and left me with no confidence. In order to take on such a daunting task as transitioning, confidence and self esteem really help. Same thing with family drama and my relationship with them. My therapist saw that was my big problem that has been causing much of my negative self perception. All in all, my gender dysphoria is only a part of my transition and the therapy I'm receiving. And I'm glad because taking on these other problems really has enabled me to move forward and progress further with my transition. Having said that, I don't know your therapist and what the sessions are like. I would just say that focusing on other topics isn't necessarily a bad thing.
By the way, not every therapist has a time frame for hrt. Mine, for example, decides along with the patient when the time is right, if it is at all. For some, that's a month and for others it's never. I'd go to someone that has experience with our community if you are worried about harsh gatekeeping.
Good luck!