Let me start by saying that your teacher is right, because of specific prompt you were asked to respond to. Besides what your teacher said about being able to take lessons about pride from many books, there's also nothing specific to the modern world or era about it. Therefore, to develop your thesis, you want to focus on the modern end of the challenge.
What is there about the modern era that fits with The Odyssey? I would say it is in the way Odysseus has to reason and trick his way through his challenges, and, especially, in how he has to adapt. This isn't a straightforward story: this is a story in which the hero changes identity and disguises himself.
Consider some of the following as you develop a thesis:
The modern world is defined by questions of identity. People self-identify according to different categories of race, gender, and sexuality. Online, they pass on other people's words as their own, and post messages under screen names. In such a world, the best, most appropriate kind of hero is one who is comfortable with change and skilled at changing. Through his travels and tests, Odysseus displays a creative ability to adapt that makes him the perfect hero for the modern age.
So it sounds like your original thesis touches on a lot of really important aspects of the epic poem, but it doesn't necessarily address the prompt. From what you've shown me, one of the most important aspects of the prompt is not the effects The Odyssey has on its readers (generally) because these could be effects that happen in any historical context, thus leading to your teacher's critique.
On the other hand, you clearly have a great understanding of The Odyssey's longevity, and how this contributes to its notoriety. I really like the second understanding that you note in your question: "The Odyssey is a myth that was passed down orally, and I think its whole purpose is to reveal human nature through a flawed but heroic character." As one of the most long-lasting narratives, one could easily argue that The Odyssey has served as the basis for all modern narratives, which all address, in some manner or another "The Hero's Journey". You could try and prove that Homer's piece is the foundation for Western literature and ideology surrounding heroism, flaws, and success.