1. They are further removed from the American cultural well. America is a branch off of the English tree, which is a branch off of the northwestern Europe tree, which is a branch off of the western Roman Empire tree. Byzantium is another sphere of Europe, and we do not get traditions from it.
2. It is a long expanse of time (1000 years) and much action to cover for teenagers when western European history has skip points. For example, how many European history classes in high school jump the gap between the end of the western Roman Empire to Charlemagne's coronation, then the Crusades, then the Renaissance. The Dark Ages gets maybe a day of talk. Forget discussing the high Middle Ages because if the Middle Ages, which had mini-Renaissance's within them, get positive press, it takes away from that liberal dream era of the Renaissance. The Byzantine Empire had many ups and downs, intrigues, wars and theological debates, which are interesting but consume time.
1. We would have to admit that the Byzantine's had a high level of culture so that the Renaissance was really just western Europe catching up. It diminishes the flowering of the West's dominance of the planet. Cathedral schooling does this with the Middle Ages, which makes a lot of people falsely assume that the Dark Ages went from the fall of Rome to the Renaissance. One dynastic marriage set up the West with some Byzantium cultural influences, which would plant seeds for the West's Renaissance, and another dynastic marriage Christianized the Russians, which lead to the Russian obsession with being the guardian of the Orthodox Church and, by extension, Constantinople (see Crimean War). Somehow Muslim copy artists of Moorish Spain get more credit for spreading ancient Roman & Greek knowledge to western Europe than the Byzantines.
2. The Byzantines were devoutly Christian and had many religious debates. God forbid that American students even pay attention to Christian history that doesn't involve killing others, expelling Jews or oppressing women.
2a. Check out the face-off between Theodosius + Ambrose (1, 2). In the western world, and possibly the entire world, it is the first instance of an autocrat committing an atrocity, lawing down the ancient law like normal, yet asking for forgiveness for his actions from another authority, the Church. There was a higher power, a higher law and a higher authority (God) to be judged by, and this was the first time an Emperor subjected himself to that judgement while on earth.
3. Women had power. Byzantine history is full of powerful women. These women are not always pure in motivation, and are not always victims. There were women who ran the Empire for their brothers, women who were cut from Lady MacBeth's cloth, and women who schemed for their children. There was no salic law preventing women from the crown in the Byzantine Empire. If cathedral teachers must teach that history is one long fight against women, how could they mention a 1000 year Empire that gave women power?
4. Byzantine history has quite a few battles with Muslims. If the cathedral is going to forever make the Crusades out to be bloodthirsty Christians viciously attacking peaceful Muslims (and good Jews), how could they cover a historical region and period that was Christian but became Muslim by the sword? Egypt was a huge Christian community and Alexandria a great source for Christian scholarship, but the Muslims changed that. How could a history class turn Americans against Christianity for barbaric behavior if it had to discuss the unnecessary & vicious "Sack of Amorium"? What about the sacking and destruction of Ani? Weird how everything the Muslims touch goes to crap? Forget that, let's focus on evil white men, schoolchildren!
5. Western Europe let them down. We had a hand in the loss of one of the world's greatest cities, Constantinople, to the Muslim Turks. Western Europe, despite Charles Martel's beating of the Moors in France, benefited from the Byzantium wars against the Saracens and Muslims throughout Eastern Europe and even in North Africa, southern Italy and Sicily. The Byzantines did this for centuries, allowing Western Europe to grow, develop and catch up to the refined world. Launching the Crusades, we weakened the Byzantines instead of recognizing them as brothers in the Christian world. The Trusk took Constantinople and challenged Austria right up to the Gates of Vienna, and we pay no attention to this. We had a hand in it.
Why would we want to discuss any of the above? America, and the West, is always on the side for freedom, justice and democracy, right? Modern America practically sprung from the skull of Zeus fully formed as to what it is now after a bump in the road we call the Civil War. History is just a never-ending procession of righteousness and progress of ever-expending liberalism and removal of oppression. People don't need religion. No society ever had religion as a positive force for decades without murdering one another relentlessly, look at Europe's post-reformation era. See all the bloodshed? What about the Crusades? Killing in the name of Christ? Pay no attention to the glorious Empire on the Bosporus. Pay no attention to the glory of the Romans that lasted another 1000 years.