what effect does people being able to legally change their sex have on the rest of society?'
Very important, yes. We need to keep foregrounding this. And the deleterious effect is not just on women's rights (which sadly many people don't care too much about), but on everyone's ability to describe and analyse reality. A society in which it is forbidden to recognise one of the most basic material truths about human beings is a society that cannot properly analyse itself, and that must move towards ever more censoriousness.
We have seen this happen on the left over the past 10 years, as trans ideology ascended and eventually become unquestionable there. There are now SO many issues on the left (especially in the US) that are completely verboten, with concomitant effects on leftists' ability to analyse and debate, well, anything.
www.vox.com/2020/7/29/21340308/david-shor-omar-wasow-speech
This article is not about trans, but about Democratic data analyst David Shor, who was fired after tweeting a study on the political effects of violent versus non-violent protests:
while one data scientist’s tweet of one political science paper should not be the last word on social movement tactics, the reasonable response to Shor would be to counter with some other form of evidence. Instead, the dialogue followed a pattern in progressive circles that often involves making evidence-free assertions about how members of various groups feel.
What role has the uncritical adoption of an ideology that claims 'people are whatever sex they say they are, and if you don't agree then you are contributing to their deaths' played in creating this type of climate?
And how can it be transformed into a healthier one (which many progressive people say they want) while 'sex change' remains a cherished shibboleth? Society cannot have free speech and open debate AND reify gender identity ideology. The latter is fundamentally at odds with the former. Because it is based on a lie.