Love and fear
Most of my biggest supporters are women so I don't believe for a moment that TERF mentality is the baseline position of independent women. Instead what I think is that the patriarchy has wounded some of them and their axe to grind is to paint transwomen with the same brush as the men who hurt them.This is basic psychology because well adjusted people don't have natural enemies and will judge others on their own merits and at face value. They don't need to make baseless general conclusions.
For example, JK Rowlings's experienced sexual and verbal abuse when young has colored her views and by playing up the angle that sexual predators could be dressing up to enter women's spaces, it clearly shows what impact it had. It may not be a reasonable conclusion (because the stats say it isn't) but that doesn't matter when it is the emotional side of the brain drawing the conclusion.
Love and fear are the governing emotions of the human condition and no amount of reason will alter either when the investment in either or both is so high.