There is a huge wave of anti-trans laws being pushed through by conservative elected officials at the state level right now. They're specifically attacking trans children and school policy, as the soft underbelly that gets conservative parents riled up for a culture war and which is legally more vulnerable.
The culture war came later, when liberals tried to use trans people to promote their own view—which you aptly characterized above—that people should be allowed to express themselves free from traditional social norms and roles. Liberals rushed to promote things like drag shows, which have nothing to do with trans people, and everything to do with voluntary self expression. As RuPaul (a cis man) put it: “Drag says 'I'm a shapeshifter, I do whatever the hell I want at any given time.’” That liberalism is what has “conservative parents riled up for a culture war,” when they mostly didn’t bat an eye at actual advances in trans rights.
By contrast, what you call the “mainstream liberal interpretation” is actually the radical position. It rejects society’s right and prerogative to impose norms and regulate voluntary self-expression. What generates concerted opposition from conservatives and moderates is the feeling that liberals are using sexual minorities as a vehicle for loosening restrictions on voluntary behavior.
You’re correct the two positions lead to the “same” outcome—but only initially. Eventually, a conflict arises between liberals, who want sanction for a broader range of voluntary self expression, and gay or trans people, who often simply want to fit in as best as they are able. You’re seeing that conflict of interest play out right now. Bostock, which granted equal civil rights to trans people, was immediately accepted by the majority of conservatives. That caused liberals to pivot to causes like drag shows involving voluntary non-conformance, which generated immediate backlash and opposition. For the reasons you explain, liberals can’t tell the difference—to them, it’s self expression, whether it’s voluntary or biologically rooted. But to everyone else that makes a huge difference.