A user left the following very apt comment on my previous answer:
You say that women have an important role in Judaism, but of course
the question was not whether women have a role at all, but whether
they have an inferior role. So the real question is, if, by a freak
accident of nature, a particular woman is inclined to carry out public
observances and governance (roles traditionally conferred upon men),
does Orthodox Judaism hold her back, and if so, how can this be
justified without considering women as inferior? – DepressedDaniel
I thank @DepressedDaniel for his serious question and offer the following reply only as a beginning, and in subjugation to the heartfelt suggestion that that any man or woman in this position (i.e., of having "freak" inclinations not in keeping with the Torah's assumptions) find and consult over the long term with a wise, sensitive, and knowledgeable rabbi, rebbetzin or mentor with whom he feels he can speak freely. The latter, in combination with (1) a peer and (2) a spiritual mentee (both of which the person in question should also endeavor to find), will provide support, comfort, encouragement, and direction in a way that one cannot obtain from within oneself. Anyone reading this post who needs help finding such people is warmly invited to email me at firstname.lastname@example.org; I will do what I can for you.
Neither G-d nor our Sages were unaware of the complexity of sexual identity, and the ways in which this complexity might trouble a binary system and its subjects. Indeed, G-d created this complexity with great deliberation and wisdom. Indeed, He created the Torah to match. Thus the rabbis identified not two but six genders--including, it seems, "the woman who doesn't seem like a woman"--and did their best to correctly determine Torah's role for each.
But these determinations would seem to do little to provide solutions for modern life, or to console the woman who looks and feels completely feminine, but has a desire in her heart to fulfill a public religious role or to perform traditionally masculine mitzvos. Shouldn't she be allowed?
Believe it or not, this question is not new, and it too has been answered by our Sages. Sefer Maharil discusses the example of Bruriah, an exceptionally gifted female Torah scholar--her words are all over the Gemara--whose involvement in Torah study ultimately led to wickedness and tragedy. It notes that
"Beruriah's bad end demonstrates her initial weakness in not relying
upon the words of our Sages. Similarly the wisest of men (King
Solomon) declared, 'I will marry many wives and not go astray' (see
Sanhedrin 21b)." (qtd. in Ellinson, Serving the Creator)
The short answer, then, is no.
The Torah, in contrast to popular wisdom, commands us not to follow our hearts. There is, Torah suggests, better life at the end of an effort to serve G-d and help others even at the expense of our own dreams, wishes, and persuasions. It is this very effort to make room for G-d that gives life to, and even creates holiness; as the Kotzker Rebbe famously said, "Where is G-d? Wherever we let Him in."
The yoke of the Torah is like a heavy load upon a strong donkey.
(Bereishis Rabba 99:9)
"Your Employer is trustworthy to pay you the
reward for your labor."
(Pirkei Avot 2.19 and 2.21)
“The reward is in
proportion to the exertion” (Pirkei Avot 5.26)
A woman who dreams and yearns with her heart to do the roles reserved for men by Judaism will no doubt feel every ounce of the weight of Torah upon her, holding her down. And yet, she is not alone in her struggle to bear the Torah. She should think of the rebuffed converts, the agunot, the incarcerated and persecuted; the desperately poor; the people whose physical or mental health makes it nearly impossible for them to keep halacha; those who cannot find anyone to marry; those who will never, because they are gay, be able to combine physical love with emotional love and life partnership within the frame of the Torah--and who keep Torah anyway. G-d gives people enormous burdens. But just as He perfectly understands these burdens, He is exquisitely aware of every calorie of energy the burdened expend trying to serve Him anyway. To paraphrase Chagigah 5b, the Holy One cries every day for one who does mitzvos in spite of a handicap. And this itself is the reward: That one serves G-d to the point of moving Him.
But the yoke bends before it ever breaks us:
"Yalsa the wife of Rav Nachman said to him, 'It is known that all that
the Torah has prohibited there is something similar to it that has
been permitted. For example, blood is prohibited while liver is
permitted,… – I want to [know what it is like to] eat meat cooked in
milk.' Rav Nachman had the cook prepare fried udder for her." Chullin
Where is G-d? Wherever we let Him in. Consider the essential roles of women as bodkaniyot, rabaniyot, teachers, principals, heads of organizations, maharatot, yoatzot,toenot, mashpi'yot, shluchot, writers, speakers, counselors, doctors for Jewish women, leaders of chesed organizations, etc., etc. Who should do these if not precisely the intelligent, idealistic women who are regretting the fact that they cannot become dayanim and chazzanim? Is it really right, and is it good for women, that the best women should be swept away from frum Judaism and into other movements? Where does that leave frum women??
That is, one not only has options to do what one yearns for: one has an imperative to do so. Take your desire to learn, to create, to inspire, and to connect with G-d through words of Torah and acts of Jewish faith, and apply them assiduously to the opportunities G-d has opened for you.
Ben Bagbag said: Turn it [Torah] over and turn it over because
everything is in it. (Pirkei Avot 5.22)
--This is how we know that anyone can live by Torah. Look at it: the Torah has seventy facets; reflected in its radiant prism, [any]one can find a life. Where one extreme seems to exist there, the other extreme is not far.
The torahitic prohibitions on women's involvement in certain acts leave, to be sure, a great deal of room to play. For example, the matters of women's learning Torah and even issuing halachic rulings have given rise to a surprisingly wide range of halachic opinions within Orthodoxy. Torah and even Talmud learning by women is permitted by various poskim--I might add the Lubavitcher Rebbe--under various circumstances. The Chid"a to Choshen Mishpat 7:12, Minchat Chinuch in halacha 78, and other sources suggest that opinions on whether a woman may pasken halacha are not a monolith. Of course, one must consult one's personal rav for a ruling appropriate to one's community. I suggest only that if this matter affects a woman's quality of life very significantly, she might, with the guidance of experts, find a way to take advantage of the Torah's nuances to find a viable path.
We are supposed to live by and for the Torah and not, ch"v, die for it. If a Jew is caused extreme suffering by the Torah's apparent view of him or her, I would first suggest that he or she has not correctly understood the Torah's view; then, I would suggest consulting a rov or a trusted spiritual counselor for advice. But pikuach nefesh--concern for human life--reigns supreme over most all we have in Torah. If you truly need to lay tefillin in order to have the koach to continue to exist, I as a woman will pasken for you: do it.
And if you don't anyway, we call that a Kiddush Hashem--a sanctification of G-d's name.