Things were still Gay & Lesbian through the late 80s, adding Gay, Lesbian, and Bisexual around that time. Then, it the early 90s, a group connected to the academic rise of Queer Theory started pushing T inclusion. It was very much a top-down move at that time and was sometimes accepted, but often resisted at the local level. Part of it was that the LGB had no idea what they were getting into. Many people, LGB or not, had never met a single trans person back then, so we had no clue what they would eventually be demanding. We just got the "poor oppressed trans people" routine, and went along with it, much like liberal feminism. For many groups, they were simply LGBT in name and had no trans members at all. It stayed that way for much of the 90s, with some notable exceptions, like the 1994 Camp Trans protests at MichFest, the first trans attempted trans incursion into female spaces. I consider that the first major shot in this conflict, unless someone knows more (or unless we count Sylvia Rivera's 1973 or so fight with Gay Liberation).
One of the most important figures in LGB activism during the 90s and 00s was Elizabeth Birch, a lesbian, who was Executive Director of the Human Rights Campaign from 1996-2004, and during her tenure, she kept the organization focused on LGB causes, refusing to let the organization be hijacked by trans activists. They hated her for this, but, at the time, they still lacked the power to do much about it. Looking back, I think Birch was prescient in recognizing the threat that the T posed to LGB communities and the achievement of LGB goals and wasn't about to make the mistakes that led to the mess we're in now.
In the early 00s, trans topics started getting trendy at academic conferences, and this gradually trickled down to places like Tumblr that drew a lot of young people into queer/trans ideology and made converts of them. And we had, as the OP noted, growing numbers of AGP TIMs who were changing the demographics of the trans population due to their victories over medical gatekeeping. There was a slow, but steady ramp-up of aggressive trans activist rhetoric during this period of the late 00s into the teens.
The real power vacuum came after the SCOTUS decision on same-sex marriage. The LGB relaxed their grip, and, all of a sudden, the T jumped in, said, "It's our turn now," and seized every position of power they could find, including at HRC.
It is a complex story. There's no one thing that happened, but the main point here I want to restate for emphasis is that at the time the LGBT alliance came into being, the LGB had no idea what they were getting themselves into. We were just going to be giving a helping hand to this poor, harmless, despised minority. Had we been able to look into the future, I think we would have said, "No thanks."