Overall, the answer is no. I think you have to be rigorous in terms of how you answer this question. If you look strictly at gay marriage, the various decisions in different countries didn't in and of themselves lead to what we are dealing with now.

The movement towards the focus on gender ideology began with academics like Judith Butler and was popularized by others including Riki Wilchins. Keep in mind that Camp Trans, the organized opposition to the female-only policy of MichFest started all the way back in 1994, and, at the time, it had little to do with LGB activism. It was aimed at feminists, not at the common enemies of LGB, namely religious conservatives and their allies.

I would argue, having been around at the time, that LGB rights and the TQ+ were fairly distinct, and it wasn't uncommon to hear from the self-described Radical Queers that gay marriage was some kind of assimilationist, bourgeois effort that wasn't part of their agenda at all. They weren't interested in monogamy and white picket fences.

What I think you have is two parallel strains that never really had that many goals in common. What did happen is that after the SCOTUS marriage decision in the U.S., a lot of LGB-founded organizations pivoted to TQ+ gender issues in order to continue to have something to do to justify the corporate money they still had coming in.

So, yes, I think you are getting the chain of causality wrong. I understand how it looks to conservatives because they aren't in a position to distinguish between the LGB and the TQ+. It all looks the same to them.