I suspect it may be a significant majority rather than a minority that feel strongly about women's rights, but until all women can speak freely with no fear of repercussions we'll never know.
I also agree with this. I work within a health care setting where transgender rights and issues are beginning to take centre stage. I am happy to support transgender people's right to access good quality healthcare and my role in this would be predominantly an educational one, ensuring that they are sufficiently informed and comfortable to proceed with treatment and as such have given informed consent to procedures based on this.
However I do not believe that transwomen are women. I believe they are different, not lesser. I would not be doing my job properly if I merely treated them as a woman, if I did so I would miss great swathes of their experience and particular issues that would potentially affect their decision making around accessing treatment.
I am not sure I could openly state the above within my workplace at present. The organisation is hyper vigilant to any perceived criticism or 'bigotry' towards transpeople and as such i would be very cautious as to who i might explore this issue with within the boundaries of my professional role.