That seems to be the right's problem with all three of sex, ascribed gender, and gender identity. Heck, even more basically than the science around those, simply understanding the difference between the three concepts seems problematic.
> And now we're moving into to the idea that race can be 'decided' by the individual.
Actually, race in the US has generally been officially been about identity and not strictly governed by ancestry or non-identity ascribed status [0] for much longer than trans rights have even been a visible issue. And, despite that, the idea that someone can have a racial identity out of line with externally ascribed race is nonetheless not remotely widely accepted on the left, despite it being the long-standing official case; see the Rachel Dolezal case.
So, the idea that transgender rights, is the leading edge of a liberal scheme to impose acceptance of trans-racial identity is deeply disconnected from the facts.
[0] except that in some contexts Native American / Alaskan Native identity requires confirmation by ascribed status from a recognized group, because it operates somewhat like a nationality rather than other races.