In your last paragraph you hit on a really important point. In many ways society is completely hung-up on gender stereotypes, so much so that, like you say, you could never pursuade people otherwise. For some trans people this can be a huge source of pain. Please, let's work together to break down these stereotypes together.
First, this is forced teaming and it's not appreciated. The issues dealt with by female people with gender stereotypes are not one and the same as those dealt with by trans people. A number of reoccuring issues for female people have been increased and made worse by the trans political lobby and agendas. Female people need to be able to talk about and deal with their own issues, be open about their own issues, and not have them blurred and lost behind issues experienced by trans people. They are two separate groups with two separate needs, voices, experiences and neither is more important than the other.
Second: it is not the job of female people to salve trans people's pain. Particularly not by subordinating their own needs and providing service to sort out other people's problems, this is the same rampant sexism that has impeded female equality and access for years. And right now issues of equality and access are more severe than they have been in decades, largely because of male people identifying as women.
Third: really the time for 'lets all work together' was about five years ago. From bitter experience of years in the trenches trying to put the brakes on the galloping roll back of women's rights, 'lets all work together' usually translates as 'shut up, step up and do all the work of putting others before yourself.' No. The time for selflessness by women and trust by women - that horse hasn't just bolted, it's died of old age.
Women are being arrested for stating biological fact and raising safeguarding concerns, being raped in prisons by convicted sex offenders, MNetters are posting here because they are rape victims who can't access women's rape crisis support, because they cannot escape dangerously abusive relationships and access women's refuges, (one MNetter was homeless for months), because they're losing accessible toilets at work - 'work together'?
No. At this point the fence mending is going to have to come entirely from those who started this, have inflicted all this harm on female people, and failed to care in the slightest about it. Women do not have the time, capacity or good will left to go fixing anyone else's issues right now.