I must say, I have never personally come across a transgender person (nor a trans rights advocate) who claims that it is currently possible to change a person's sex, though perhaps you have. The claim is of a change in gender, which is perfectly in line with the modern scientific understanding of how sex and gender function (i.e. sex being biological, gender being socio-cultural). This is why the American Psychological Association says that "incongruence between one’s sex and gender is neither pathological nor a mental health disorder." Transgenderism is being recognized fundamentally as the incongruence between sex and gender, not a change in sex (which is how conservatives mistakenly understand it).
Perhaps there is also something of a generational element to this issue; for instance, I've noticed that the term "transexual" has mostly fallen out of favor with younger trans people (I met a great many trans people, and none of them use the term), whilst older trans people seem more likely to use it. I wonder whether this has something to do with what you say about the older members of your support groups seeming to place more emphasis on the biological aspect (unless I misread what you said).