I think the straightforward answer to this question of who is behind trans lobbying is simply "men are".
As women we are taken aback by the speedy progress made by the trans movement because it is new for us to experience a men's rights movement which is going after something men don't already have.
Men run society and dominate on all sociopolitical levels so when they want change they don't have to protest from outside power circles the way that women and real minorities do. When men want change they lobby from within power circles and they lobby from a position of high status - they don't have to start their protest with time and energy wasting pleas to be considered as fully human and legitimate in the way that women have to.
Once you add in the fact that this men's rights movement is deeply anti woman and women's rights + a money spinner + a jolly interesting medical experiment (if you are that way inclined and much of the medical industry is) + numerous other fucked up factors that I haven't thought about and it's no wonder that the whole thing has become a powerful and fast moving juggernaut.
Plus we are currently in a neoliberal, postmodern, identity politics dominated, post critical thinking social context.
Plus the role of an increasingly messed up human sexuality as influenced by porn culture, rape culture, pedophile culture, objectification of women and fetishisation of our body parts.
Plus the internet and the huge impact it has on both movements and individuals.
Plus, and I think this is not much talked about but hugely important factor; men's (as a class) God complex and hatred of women for being the ones who carry and birth babies. (And their desire to take that ability away from us and at least control it themselves if not have it themselves; see history, religion, gendercide, abortion and reproductive rights, the institutions of marriage, motherhood and virginity, medical reproductive interventions, etc.)