The same halacha that prohibits a woman from holding public office, also prohibits a woman from even being the President of a Shul. A woman is essentially prohibited m’doraisa from holding a position of authority. This law comes from Tanach.

And this point was upheld by Rav Moshe, Rav Kook, Rav Yoshe Ber Soloveitchek, etc. etc. Young Israel has a strict rule against member synagogues from having a female president. There is no rabbinic authority who is anywhere near the stature of even one of those upholding this fundamental principle that disagree with it.

As far as Rav Kook’s wishes not being followed in the State of Israel, that is simply because the State of Israel is a secular non-Jewish state, whose secular leaders do not take direction from Torah leaders. They violate the Torah across the board, so why should this be different? The decision to allow women to vote and hold office in Israel despite Rav Kook’s strong objections was made by the secular Zionist leadership, who don’t keep the Torah, not by any rabbinical leaders to accept another position. The government violates the Torah in public policy on many levels.

The question of whether woman should vote is a somewhat separate discussion then running for office, that rabbonim have come to different conclusions on (voting). But even the rabbonim who l’chatchila would not allow even to vote, by and large decided that, b’shas hadchak, since surrendering our women’s vote on principle would effectively reduce our influence by half since the rest of the world will allow them to vote, and since women voting isn’t equivalent to a woman holding office with authority, they permitted that.

The fact that the woman was created from the man’s rib seems to suggest the male’s superiority over the female. This is the understanding, for example, of Ralbag:

“Woman was created from man, because he is the reason for her existence, that is to say, she was created to serve him… She was created from him, so that she should be more obedient to him and perform the services that he requires… This is not the case regarding other living creatures. And for this reason they were created together from one place.” (Ralbag, Commentary to the Torah)

At the end of the passage, Ralbag argues that it is only in the human species that the female is inferior to the male. Hence, it was only the man who was created alone without his wife, she being created only afterwards. [This explanation is also found in Ra’avad’s “Ba’alei HaNefesh,”]

The Ba’alei HaTosafos explained that it was because of her inferiority and subservience to her husband that the woman was created from his rib:

“There is a difficulty: why was the woman created from a rib, and not some other organ? So that she should be bent at the ribs and subservient to her husband.” (Ba’alei HaTosafos Al HaTorah)