Before I elaborate on my question, I must first direct attention to this clip where Ayn Rand flatly states that she wouldn't vote for a female president. "If we had fallen that low, I might." She stated in that interview that she had written an article giving her reasons as to why she took such a position. I searched for it and could not find it. I was wondering if anyone can help me either find that article or simply tell me what Ayn Rand was trying to say.
To present the idea that the essence of femininity is "hero-worship" can come off as a bit contradictory. Yes, Dagny Taggart is a hero-worshiper to John Galt in Atlas Shrugged, but she did state to James' newly wed wife Cherryl Brooks, "I'm the man" at the wedding. Dagny was trying to be a man to whom others can worship. To say that to Cherryl Brooks would imply that she would like Cherryl to admire her, regardless of gender. It would seem that hero-worship has more to do with a person's integrity and less about a person's sex. Why would it necessarily be a bad thing if a female is "in charge of the military," as Ayn Rand said?
The video linked in the question is from an appearance by Ayn Rand on The Phil Donahue Show. The article of hers that she refers to is titled, "About a Woman President" and was originally published in the December 1968 issue of her magazine, The Objectivist. The article was later republished in its entirety in the book, The Voice of Reason (VOR), Chapter 26. Two key paragraphs from that article are excerpted in the topic of "Femininity" in The Ayn Rand Lexicon.
Because of the psychological issue of femininity and what a position such as President of the U.S. would do to it, Ayn Rand concluded her article as follows:
For a woman to seek or desire the presidency is, in fact, so terrible a prospect of spiritual self-immolation that the woman who would seek it is psychologically unworthy of the job.
If one accepts Ayn Rand's view of femininity, the logic of her position on a woman president seems to follow naturally. But her view of femininity has been highly controversial over the years. Another article that discusses feminine hero-worship and romantic (sexual) love is "Of Living Death" in VOR Chapter 8.
The "Woman President" article emphasizes that Ayn Rand is referring only to the highest political office, such as U.S. President or foreign heads of state, not to lesser offices such as congresswoman, senator, judge, "or any similar rank." She is also not referring to non-political positions of leadership, such as "a woman who heads a business concern." To understand Ayn Rand's reasoning more fully, one will need to study the article (along with heroines such as Dagny Taggart in Ayn Rand's fiction, and various other non-fiction discussions such as VOR Chap. 8.)
One memorable excerpt from VOR Chap. 8 is the following, in response to a substantial excerpt from the Papal encyclical, Humanae Vitae:
I cannot conceive of a rational woman who does not want to be precisely an instrument of her husband's selfish enjoyment. I cannot conceive of what would have to be the mental state of a woman who could desire or accept the position of having a husband who does not derive any selfish enjoyment from sleeping with her. I cannot conceive of anyone, male or female, capable of believing that sexual enjoyment would destroy a husband's love and respect for his wife—but regarding her as a brood-mare and himself as a stud, would cause him to love and respect her.
Refer also to the topic of "Sex" in the Lexicon.
The question also states that when Dagny said, "I'm the man":
Dagny was trying to be a man to whom others can worship.
That's not how I understand Dagny's statement. Here is the context leading up to Dagny's remark (Part II Chapter II, p. 374 in my Signet paperback edition of Atlas Shrugged):
"There's something I want you to know," said Cherryl, her voice taut and harsh, "so that there won't be any pretending about it. I'm not going to put on the sweet relative act. I know what you've done to Jim and how you've made him miserable all his life. I'm going to protect him against you. I'll put you in your place. I'm Mrs. Taggart. I'm the woman in this family now."
Dagny is simply stating that Jim has been of little actual use in the running of Taggart Transcontinental, and that Dagny has been the actual driving force behind it, despite the repeated public credit that Jim has tried to claim. Cherryl hadn't figured that out yet at that point in the story, but she eventually does discover it.