There is sufficient motive
A female writer would have a clear need for a pseudonym like William Shakespeare.
Shakespeare draws on a number of existing sources. But his female characters and friendships are rarely lifted from them. When he includes historical sources, he also, at times, feminizes them. (Plutarch and Anthony and Cleopatra). Shakespeare also builds authentic female relationships and characters at a time when no other masculine playwrights could.
Many of the heroines Shakespeare creates refuse to play by the rules. 10 of his female characters defy their fathers, eight disguise themselves as men and six lead armies. These bucked the trends of the day. Beatrice, for example, in Much Ado About Nothing, is infuriated by the limitations of her sex. Kate in the Taming of the Shrew will not be silenced and controlled by her husband. Emilia has strong opinions on women's equality in Othello. The most intriguing relationships that are developed in Shakespeare's works are often between women (Paulina and Hermione in a Winter's Tale; the Merry Wives of Windsor, and many more). How was Shakespeare able to write with such a unique perspective on women's issues and relationships? Because he was a woman.
Firstly, female authors do not hold a monopoly over female themes and seminal feminist works. You only have to look at Gustav Flaubert and George Bernard Shaw's works to see that male authors can explore feminine themes with as much intricacy and sensitivity as female authors. Therefore, Shakespeare's dealing with the limits of the female sex and challenging the role of women in society is not in itself evidence of the author's gender. Secondly, Shakespeare is able to develop such strong female relationships because he has a masterful understanding of the human condition. He was able to construct deep and meaningful relationships among many of his characters, not only his female characters. The relationship between Lear and his fool, Hamlet and Horatio, and Caesar and Brutus are also masterfully crafted. His ability to build convincing and intricate female to female relationships is not an indication of Shakespeare's gender but of his genius.
[P1] Shakespeare's works explore feminine themes in a way none of his contemporaries could. [P2] His female characters and relationships are also among the most intricate and developed. [P3] Therefore, Shakespeare was likely a woman.
[Rejecting P2] His male to male relationships were just as well crafted. [Rejecting P3] This is evidence of Shakespeare's literary intelligence, not his gender.