The thing is, this is nothing to do with the interests of trans men and everything about removing 'women' as a term for female people. If it were truly about the best possible cancer care, this attack would encompass the messages around men's care, too. After all, transwomen retain male biology. They are affected by public health messaging around male cancers. Yet there has not been a single effort to force the words to change around prostate cancer. Why? Because focusing on language there would not be an attack on women's rights to define ourselves, that's why.
If this were about cancer care, then the same pressure would be applied to the word 'man'. It's not. Because this is about removing any way for women to talk about our own lives, and experiences, and how our bodies impact the way we are treated in this world. And at a time where rape has been effectively decriminalised, porn culture is everywhere, women are made redundant in pregnancy as a commonplace and the reform promised on that since 2017 is yet to happen, and a girl is raped on school premises every day of the school year... then that's rather a major fucking problem. And how can you name that problem, if you aren't allowed to name the group affected in any but the most dehumanising of ways?
Apart from anything else, in terms of cancer care half of all women don't know that they have a cervix, but every trans man and non-binary female knows that they have a woman's biology. So if the real concern was the noble desire to save 'even a single life' then they would want the word woman used, to protect all individuals with a cervix who don't actually know what a cervix is. Using the language that best communicates accurate meaning to the largest number of people saves lives. They don't want or care about that, because it's just part of the same concerted effort to detach the very language to describe women, as individuals and as a collective, from women, so that men can insist upon denying us any workable, definable, non-dehumanising category of our own.
This has nothing to do with improved engagement with cancer screening, and everything to do with misogyny.