Whatever the reasons for covering hair (e.g. fear of heaven), they seem to equally apply to men and women. So why don't unmarried women have to cover their hair?
Let me break this question down. First, there is a minhag (custom) that men should cover their head as a sign of reverence to G-d. The custom was codified as halacha for men (Orach Chaim 91:3) which stated that it is forbidden to say G-d's name or to even walk into a Synagogue with your head uncovered. For me the practical aspects are (a) that the kippah serves as reminder that I stand before G-d, and (b) that covering my head stands as a signal that I am a Torah Jew and that should I stray from the Torah my behavior will be seen and noted by others and reflect poorly on the Jewish people.
Married women cover their hair, and not just their heads, for a different purpose. Their hair covering serves as a shield to hide one of their most attractive features to men other than their husbands, and also states clearly that the woman is married and not available.
So to your question: Why don't woman cover their heads as a reminder that they stand before G-d? The Tzitz Eliezer (12:13), citing numerous sources, says that there is no logical legal basis for unmarried girls to not cover their head when saying Hashem's name or going into Shul. He even brings from the Ish Matzliach who says that in Tunisia they enforced this Halacha and all girls covered their head when men would also be required to do so by Halacha. So why is that not true anywhere else?
You, in your answer, hit on one reason. The Chasam Sofer, commenting on Nedarim 30b, and the Taz (OC 8:3) stated that in Talmudic times men often didn't cover their heads, but we adopted the custom because non-Jews prayed with their heads uncovered, but required their women to cover their heads, and we want to make sure that we do not follow their customs.
Another reason follows the reasoning as to why men are required to daven with a minyan or wear tzitzit and women are not. Men start life at a much lower spiritual level than women (Sotah 11b, Niddah 31). Similarly, the Maharal explains that men require more help to overcome their inherent aggressiveness in contrast to women (Hidushei Aggadot I, Kol Kitvei Maharal).
Rabbi Samson Raphael Hirsch in his commentary on Genesis 17:14, summed it up beautifully:
"The pure feminine sex, if it descends from Sarah, does not require the external sign of the covenant with Sha-dai, the God who "sets the measure." It itself bears this warning of "Dai" ["enough"] within itself, in the pure feeling of the limits set by its tzniyus with which the true Jewish women are filled. She has the tendency by itself to submit herself to all the laws of purity and godliness, and demands such submission from all that come into contact with her."
Hence, women don't need to cover their heads to remind themselves they stand before G-d, but men need all the help they can get.
Qitzur Shulhhan 'Arukh - Yalqut Yosef (Even HaEzer 21:9) writes (my translation):
פשט המנהג שבנות רווקות הולכות בגילוי ראש ברשות הרבים, שמעיקר ההלכה אשה שאינה נשואה אינה חייבת בכיסוי ראש. ורק בעת שמתפללות או מברכות ומזכירות שם שמים, תכסנה ראשן
The general custom is for single women to go in public with uncovered hair; because, per 'Iqar HaHalakhah, an unmarried woman is not obligated to cover her hair. Nevertheless, when praying, reciting blessings or reciting the name of HaShem, they should cover their hair.
While on this topic, there is a tradition of unmarried women and girls covering the head or covering the hair, but it is not the Ashkenazi custom.
There is a ruling in the Shulchan Aruch that all women, married and unmarried, should cover the head / hair, but "unmarried" was interpreted differently by different commentators. Some said that it does refer to all women and girls, others to both married women and women who were once married.
If you consider that Karo and Maimonides lived in Muslim societies, where all girls take up hijab at puberty, Jewish girls would be living in a society where basic dress meant covering the head and the body with a long veil whenever she went out.
Shortly after puberty, a girl would be married or engaged. So perhaps Karo did not feel that the statement "married or unmarried" required much explanation.
In contrast, Ashkenazi commentators lived in Christian societies, where men would bare their heads in church while women covered theirs, in accordance with the NT. There may have been a desire to differentiate Jewish practice from this.
If you go to a traditional Yeminite family, you will see women and girls cover their heads when saying brachot even if they do not cover the rest of the time. (They will even use a kippah if one is handy.)
Yeminite customs are some of the most ancient and unchanged from Biblical times and I think this hints at a tradition of all women (like men) covering the head when pronouncing the name of G-d.
However, since women also cover the head for modesty, there is a lot going on here. And that is why you can get 10 different but still legitimate answers to this question.
Maybe not a direct answer but it could help:
The Chazzon Ish is quoted as saying that the extent of a woman’s hair covering is the extent of her yiras shomayim. Pe’er Hador 3:page 18. When this mitzvah (commandment) is kept properly and with the correct attitude it imparts considerable Yirat Shamayim (fear of heaven) to the person. Men cover their head with a yarmulke (kippah) or hat in line with the recommendation of Chazal (our Sages), "Cover your head so that you shall experience the fear of Heaven" (Shabbat 156b). If covering just part of the head as is practiced by men has such an effect, how much more must covering the complete head have a deep and far reaching effect on a woman's Yirat Shamayim. Considering that the head is the most distinguished and most significant part of the human body as Chazal say, "The head is king over all the limbs" (Shabbat 61a), the influence of kedusha that is transmitted to the whole person by a mitzvah done continuously with the head must be immeasurable. Significantly, Chazal say that when a man wears tefillin (phylacteries) he has a special defense against serious sin (Menachot 43b). Accordingly, a woman who lacks the outstanding mitzvah of kisuy saarot (hair covering) because she does not fulfill its halachic requirements, withholds fromherself a vital source of spirituality and inspiration. "While a mitzvah is being carried out it shields and safeguards the person" (Sotah 21a).
So like you said covering the haid (e.g. fear of heaven), seems to equally apply to men and women.
I have been told in the name of the Chasam Sofer that the only reason we cover our hair is because not to copy the goyim. In church only men go bare headed that is why only men have to cover their hair. This would really only explain why they cover it in shul but not why men cover their hair in the street or at home.