I remember a guy on this site who used Kaiser, and I knew a guy in town who did the same. They both wound up seeing a therapist only once a month. That's great if you need only a T letter after three months of therapy and you don't really have any other issues. It's not so great if you actually have other issues to deal with and want to see someone once a week.
I don't know whether some pushing would have gotten these guys more frequent therapy; I think both of them just wanted letters (or approval to start T). The thing is, the guy I knew in person really did seem pretty messed up to me, but his issues went unaddressed, and he stopped therapy after he got his approval for hormones. I'm not a doctor or therapist, but I wouldn't have given this guy a T letter on the fourth visit. I suppose he could have held back in therapy so that the therapist would think he was more grounded than he actually was, but isn't the therapist supposed to sniff out stuff like that? Of course, I have since lost track of this trans guy, so I don't know how he is doing now. He got his T, and he could be doing fine right now.
I know that my feedback is only secondhand, so I hope you get some more definitive replies from people who are actually Kaiser members. Do you feel that you need counseling, or just letters/approval for hormones?