I don't think it's fair to say the Sun is the all-encompassing personality, or else we'd be falling victim to what is wrong with Sun sign astrology. If the Moon is your emotions and your emotional responses to things, I would think it's just as if not more important than the Sun. The Moon is a very personal and deep part of the personality.
Regarding this addition: "So for example, a woman who has her sun sign in capricorn has a father who has capricorn qualities and she looks for these qualities in his husband too (nice to know that this is scientifical prooven!)
But because it is the sun in this sign, she wants to fuse with his husband, she can't see herself disconnected. And that's the reason why woman have more difficulties then a man when seperating. Because men go further with their lives (their wifes are moon - related, so their personality is not affected), but women suffer more, loosing the sun is like loosing a father, loosing herself. Not knowing what is going to happen to her after the separation.."
In a sociological perspective, a woman looks for traits of her father in men (if she is heterosexual) because this was her example of what a man/husband/partner is to be like. I think women on average have a harder time separating at times because all in all we are taught to consider marriage/partnership as the end goal in life, and when it does not work out we are more crushed than the man is, because we feel that we have failed at our primary life goal. Also, if the man was the primary breadwinner, and/or takes a lot of the assets when they divorce/separate, and/or does not take responsibility for any children when they separate, the woman is undergoing very significant financial stress. Men generally go further in their lives because they are encouraged to pursue material goals and women are generally encouraged to pursue relationships with men and to bear children. It doesn't have to be this way.
Personally I don't think astrology is naturally sexist, but interpretations can be. Also, our natal charts are snapshots of the moment we are born, born in a sexist environment. If I have stressful aspects relating to relationships (and I do), I consider it to be a result of my environment and my family history, not of the universe's grand scheme to make women subservient. Perhaps there is a lesson here to acknowledge this inequity and to do something to change society, and to change for the better in upcoming generations and end the cycle of abuse.
Regarding gender, sex, and binaries, I always view this topic as sex is sex and gender is cultural. We're expected to behave (gender) a certain way based on our sex. The expectations vary per culture but worldwide the female sex is abused for their reproductive capabilities and for sexual access to their bodies. Part of the abuse is through the vehicle of gender - all designed to enforce submission. Think restrictive clothes and submissive postures of models in women's magazines for Western cultures. When naturally, our actual personalities deviate from these expectations but are repressed by culture. Astrologically, perhaps something like an afflicted Sun and/or Moon would indicate the struggle to express oneself as one truly believes one is and not from the expectations of others. I have my Moon inconjunct my MC and I have seen it said this makes one a "people pleaser," for example.
In short, it is two-pronged. Women are not naturally submissive, but because we are generally taught and sometimes forced to be, we can be. This is the self-fulfilling prophecy. It takes some self-awareness to break free.