Hannah Arendt taught that when language is twisted so that black is now white and white is now black, all meaning is lost.
A society that removes the power of the truth from the people can do anything to the people without being challenged.
The use of pronouns is a method to control people and remove the power of being able to speak the truth.
If you can not define woman because what you understand to be woman has become fuzzy or you are afraid of identifying whether someone is a woman you remove the power to express what problems a woman might uniquely face.
Thus pronouns are important as they blur the lines of communication and they allow errors to be possible and for abuses of power against women to take place without being seen as clearly as they should be.
Healthcare relies on understanding the sex of someone to give good and appropriate care. In a busy hospital, a doctor or nurse should check notes every time but in reality this doesn't always happen. A patient always referred to by certain pronouns might slip through the net.
Thus it is important to a doctor to maintain pronouns based on sex for the patients safety and the doctors own protection from error or the errors of others.
It helps to preserve the integrity of data so that for example you know that the blood of a male isn't included in a clinical study which is only for women's related issues.
Pronouns are not neutral words which are only about being polite. They are a way of identifying sex and conveying other meaning to others.
A male using a female pronoun is exercising power over women by demonstrating that women can not define themselves on the basis of sex. This leaves women with less power.
The desire for a woman to preserve her dignity and right to privacy for example on a ward from the male gaze is rendered invisible. She comes second to the rights of the person identifying as female and using female pronouns.
It removes the power of a woman to consent to with full information. It has the potential to leave vulnerable women with a history of abuse of some description in situations where they feel uncomfortable and find it difficult to challenge for whatever reason.
There are many reasons why pronouns should not be seen in the context of merely a way of being polite. The use of pronouns is much more than that we should understand that pronouns convey a huge amount more as a communication tool - they are identifiers and markers of power.