One of my fellow teachers who is new to classical Christian education asked this ⬆️ great question a day or two ago. She also asked why we study Latin. If we studied the Greek language, at least the kids could read the New Testament, right?
 
I've heard similar questions before, and realized that I did not have a ready response.

Why Latin?

To answer the Latin question first, there are many reasons for it:

• On a practical note, I believe something like 50% of English words are rooted in Latin, helping us better understand words that we don't know in our own language. I believe Greek makes up another 20%.
• Learning Latin makes it much easier to learn the romance languages (Spanish, Italian, French, etc), or from what I've heard, any language.
• Learning Latin allows us to read original sources. This serves multiple purposes:
  • Some ideas can be communicated in one language, but not another because the vocabulary just does not exist.
  • There can always be errors and omissions in translation (unintentional or not).
  • Many historic texts have not been translated.
  • C.S. Lewis said that for every new book we read, we should read 2 old books. G.K. Chesterton said "Tradition means giving votes to the most obscure of all classes, our ancestors. It is the democracy of the dead. Tradition refuses to submit to the small and arrogant oligarchy of those who merely happen to be walking about." We could say the same about reading old books.
• The majority of writing by church fathers is in Latin. Some might argue "but we're not Catholic!" However, all Christian churches descend from the Catholic Church. The Protestant reformation addressed many issues, but even the reformers looked back to earlier church fathers with admiration.
• Most of the historic work in science and academia has occurred in Latin.
• Latin is a "dead language," which means we don't have to deal with the pesky nature of living languages to change meanings of words, or grammatical rules.
• I've heard some argue that regardless of any "practical" benefits to learning Latin, we should learn it simply for the joy and beauty of it.

Why not Greek?

As for learning Greek, many of the same benefits would apply, with the added benefit of being able to read the New Testament in its original language. While I expect that most classical schools prioritize Latin over Greek, I would expect it to be a close second. At our school, we hope for our students (and teachers 🙃) to become fluent in both Greek and Latin.

Why pagan Greek myths?

Though I feel I have a decent response for why study Latin and Greek, the question about studying Greek Myths still had me stumped.

I could say something about "all Greeks were pagans (they were pre-Christ), and yet Paul quoted them." In Acts 17:28, Paul quotes Epimenides when saying "In him we live and move and have our being." The original quote would  have applied to the pantheon of Greek gods, but Paul applies this to Christ; he baptized the pagan culture around him with the Gospel. Our kids don't know anything of the pagan Greeks though (we're the ones teaching it to them) so that argument would break down.

I could  also point to Socrates who seems to have been a monotheist in a world of many gods, or Heraclitus who spoke of a divine Logos. Both lived prior to Christ, but were seeking truth and had they lived a few centuries later, might have found it in Him.

After our conversation, I did some digging. The following can be found in the front of Memoria Press' Greek curriculum (emphasis mine).

Next to the Bible, Greek myths are the most important set of stories in our history. Greek myths are universal stories that describe human character types and the human situation in a wonderfully imaginative and entertaining way. They are everywhere in English literature and in our culture, from the names of cars and shoes to the names of constellations and planets.

Except for the Hebrews, all ancient cultures possessed a mythology. The Greeks developed the most extensive, imaginative, and literary mythology in the Western world. Many ancient mythologies had dark and evil gods like the Baal religion in the Old Testament, the religion that Phoenicia and Carthage had spread throughout the Mediterranean world.

The Greeks had the good sense to create their gods in the image of man and give them the better, if sometimes comical, attributes of the human race. The Greek gods, being quite human themselves, detested human sacrifice. As a literary creation, Greek mythology provides us with symbols and metaphors for human types, attributes, and ideals. Greek mythology is universal in our culture, because like Latin and Greek, artists, writers, musicians, and scientists have mined it extensively for two millennia.

Greek myths and other ancient myths and legends often have similarities to Bible stories. These parallels provide good opportunities for distinguishing between the Bible and other ancient writings. The fashion today is to lump Scripture with all other ancient writings as unreliable myth and legend.

For instance, the human race seems to have a collective memory of a universal flood and a "Garden of Eden" since almost all ancient mythologies have these two stories. This can be taken as nearly conclusive evidence for the opposite conclusions that (1) these two "myths," in fact, refer to things that are really true or (2) the universality of these stories shows that they are merely myths.

One way to approach this with your students is to ask them what they would think if these stories were found only in Scripture and nowhere else. The skeptic could then say that their uniqueness is conclusive evidence for their unreliability since if they were actually true, other cultures would have also had myths about such universal events. So "evidence" can be used to support whatever position one wants to believe. Students who have gone through these thought processes will be prepared for the mythologizing of the Bible that will confront them later in life.

A mythology is a culture's attempt to supply answers to the mysteries of human life. Man uses his imagination to penetrate beyond what his senses and reason can tell him. The Greeks in both their mythology and philosophy represent man's best efforts to penetrate to the divine. The Greeks show the limitations of the unaided human mind and, by doing so, witness to the divine origin of Scripture. The Greeks who created gods in the image of man, prepared the way for the God who created man in His own image.

Also, I thought this was a good response to the question (emphasis mine).

One thing that I consider important to keep in mind as you teach mythology is that while they need to know the references for future reading of the classics, the Greek gods give insight into the human condition. They were personifications of virtue and vice that the Greeks observed. For example, when speaking about Ares, the D'Aulaires book says, "Ares, god of war, was tall and handsome but vain, and as cruel as his brother Hephaestus was kind." The student should be led to see that war can be attractive (tall and handsome) for what a people can gain (land, resources, and hopefully peace from enemies in the future) but that it is motivated by vanity (a country always thinks it is better than the other) and leads to tremendous cruelty. And the contrast to Hephaestus, the god of the forge, is telling as well: men who spend their days occupied in honest work can more easily see the needs of others and empathize with them but those like Ares who stand idle react like him: "When Ares heard the clashing of arms, he grinned with glee, put on his gleaming helmet, and leapt into his war chariot."

It can easily be overlooked that the Greeks were the first to develop the idea of human virtue (habits to do the good) and what that meant. They listed four primary virtues: prudence, justice, fortitude, and temperance. Early Christian apologists would take those virtues and promote them as important for all people, whether Christian or not. However, only with the revelation of Christ did we receive the idea of the theological virtues of faith, hope, and love. All this relates to mythology when you realize that the myths were the instruments by which the Greek people learned their morality. That they had a god of war does not mean that they worshiped war, but that they were aware of how serious a decision that was and that the outcome of a war was out of their hands and they may be the loser--so they must tread carefully when considering it. That they had a goddess of beauty does not mean that they were full only of immorality and licentiousness (obviously as sinners there was some of that too) but that there was an awareness of the goodness of creation (cf. Genesis 1) and it can be a calling to lift our minds to more heavenly things (cf. Col 3:1-2). So some important lessons should be learned from the mythology if understood in the proper way.

I think your daughter is somewhat correct in characterizing the mythology as "dark"--after all, the light of Christ had not been revealed yet. Christianity's saving message completely changes our outlook while the Greeks were left alone only with their human talents--which we all know will not go far without grace. Contrasted in this light, it becomes easier to see the line between the human and the divine and the need for grace becomes clearer.

As I said above, it is necessary that the student know the references they will come upon in reading classic literature. Dante, Keats, Milton, and Shakespeare (just to name a few) all weave references to mythology throughout their works. Just to give you an example, in "A Midsummer Night's Dream," Puck (a fairy) attempts to play Cupid (the Roman god of Love; his Greek name was Eros). That is only clear by the words of Puck that reference Athens, "Through the forest have I gone / But Athenian found I none..." (cf. MND 2.2.66-83). Cupid is only mentioned by name one act later, and if you didn't know that Venus is Cupid's mother, you would not understand what these words mean: "When thou wak'st, if she be by, / Beg of her for remedy" (idem 3.2.108-109). Instead of thinking of the Roman goddess, you would be thinking of the planet! And as you may have noticed, Shakespeare will use the Greek and Roman equivalents interchangeably. A student's ease in recognizing the Greek and Roman mythological figures will help tremendously as they read literature. 

Now on to Latin: a week ago I was in Nashville for a homeschool convention and I was speaking to a woman who tutors a several children. After she told me her experience, I told her I would love to have her speak at conventions for me! She said it was so clear in her students how Latin helps put structure in their thoughts. She had a clear view because she dealt with students who didn't have Latin, students that did, and students that were learning. Now I wish I had recorded her...

I can only attest to my own experience growing up--I started learning Latin in 3rd grade and continued studying in partly through college without a break. Learning four other languages was not difficult compared to what my peers went through who had not had Latin (even those that grew up bilingual). As for the structure of my thoughts, I have no way to compare that to those around me! I have the same issue when it comes to students that I have taught. Every student I have ever taught has also been studying Latin. I have no "control group" to which I can compare my students. I will let the mothers who have implemented Latin at home give you their own testimony – which I am very interested in hearing as well.

Lastly, tonight while looking for something else, I was lead to Augustine's "On Christian Doctrine", Book II, Chapter 18.

But whether the fact is as Varro has related, or is not so, still we ought not to give up music because of the superstition of the heathen, if we can derive anything from it that is of use for the understanding of Holy Scripture; nor does it follow that we must busy ourselves with their theatrical trumpery because we enter upon an investigation about harps and other instruments, that may help us to lay hold upon spiritual things. For we ought not to refuse to learn letters because they say that Mercury discovered them; nor because they have dedicated temples to Justice and Virtue, and prefer to worship in the form of stones things that ought to have their place in the heart, ought we on that account to forsake justice and virtue. Nay, but let every good and true Christian understand that wherever truth may be found, it belongs to his Master; and while he recognizes and acknowledges the truth, even in their religious literature, let him reject the figments of superstition, and let him grieve over and avoid men who, "when they knew God, glorified him not as God, neither were thankful; but became vain in their imaginations, and their foolish heart was darkened. Professing themselves to be wise, they became fools, and changed the glory of the incorruptible God into an image made like to corruptible man, and to birds, and four-footed beasts, and creeping things."

In summary, the Greeks did identify some truth, even if that was simply man's depravity. All truth is God's truth. We should point out the truth, and reject the lie ("figments of superstition"). As our students see that the Biblical account is the "myth made fact", "Perfect Myth and Perfect Fact," this will steel our students against the world's attempts to mythologize Scripture and treat it as just as fictitious as the rest.