I've been lurking on this site for a while, but I've not posted...til now...bwahahahaa.
Seriously, I think that while some therapists are not very good at the job, some are quite excellent and "therapy" has gotten a lot of undue bashing.
The standard way to get access to T (and surgery) is through therapy. If you're going to a therapist, why not take the time to make it productive?
I didn't *need* therapy. I wasn't depressed. I wasn't confused. But I told my therapist on day one, "This is a process, and I'm going to take it seriously," because....why NOT? (And I knew my gender therapist was one of the "good" ones because her attitude was "I'm not here to stop you or be a gate-keeper, even if that's how the SOC may seem.")
A good therapist will get to know you, probably have some insights about the process that you aren't aware of (from having treated many people who are not you), and help you figure out what stuff you want to talk to about or think about. A good therapist will get to know you and be in a position to help you communicate better with your mom. You are asking a bunch of strangers on the internet for tips, when that is exactly the kind of dialogue a therapist can have with you. A good therapist will get to know you well enough to recommend to your Mom that you be allowed to take, at the minimum, hormone blockers NOW. Totally reversible. Part of the recommended treatment of minors. AND it will improve the effects when you do start taking T. And on and on.
The way I see it, you are asking: how can I get my Mom to let me see when a therapist when *I* view a therapist as totally worthless except for the magical letter/piece of paper (which your Mom right now is uncomfortable with you having)?
The obvious answer is: start believing that you may actually get something useful and productive out of therapy and then you might convince someone else of it.
Tell your Mom that a therapist will help you explore the options available for you as someone with GID and that you are interested in talking to someone who can help you who is trained to do this stuff, and actually MEAN it, and she may go along with that.
FWIW - I'm not a minor and my situation is very different. However, while I was initially skeptical of therapy (and had seen a fairly useless counselor once before in my life), I feel like I do get something out of my therapy sessions. I'm still not depressed or confused or suffering. We talk about all sorts of things - professional, personal, transitioning, treatment of TS people or issues more generally. She lends me books or resources that she knows would be interesting to me. She gives me writing "homework" that gets me to express myself in a way that I don't in talking. If I feel like ranting about something totally irrelevant, I can do that there. In general, though, we laugh a lot. I've learned a lot about myself and how I feel about being trans in this world through this process, even though I didn't *need* it to be worthy of T or surgery.