The short answer would be, if you don't like it, don't read it, it's sure no skin off my nose.
The reasonable answer would be that such authors are assigned because of the enduring value of their prose and poetry. By and large, Poe's poetry is all that is offered, although his short fiction is among the best in the English language.
The Fall of the House of Usher is the "perfect" short story. There is nothing, not a single line, in that story which is not directed to the climax of the story. His other short fiction was innovative, too--
The Murders in the Rue Morgue and
The Purloined Letter are among the earliest examples of "crime" fiction.
The Gold Bug evokes a gothic mood which has the most important, and typical, of the attributes of Poe's fiction--surreality. It is also a story of buried treasure written long before Robert Louis Stevenson's
Treasure Island. Discovering what is actually real, what the truth is, lies at the heart of his short fiction, whether it is actual detective fiction, or other types of stories. Either the reader cannot discern immediately the reality, or the protagonist of the story has lost touch or is losing touch with reality.
The Premature Burial,
The Tell-tale Heart and
The Pit and the Pendulum are the earliest examples of "horror" stories. Poe was extremely inventive, and that was not appreciated in America in his time, and it was only after the French poet Charles Baudelaire "discovered" Poe a few years after the latter's death, and translated what he considered the best of his short fiction in a compilation entitled
Histoires Extraordinaires that he became well-known in Europe. Today, he is better appreciated in Europe than in the the United States, where it is chiefly his poetry which is taught (not a bad thing, his poetry is quite good), but the short fiction is either ignored, or the crucial aspects of the psychology of his stories is glossed over. The titles in blue above are "clickable" and all lead to a site at which his entire body of work may be accessed.
Samuel Langhorne Clemens, known to the world as Mark Twain remains to this day my favorite American author. I cannot recommend a reading list for you, because i like everything i've ever read by him. It is odd that you have mentioned these two authors, as both were iconoclasts. Poe was a rebel against the Protestant Ascendancy in America and its pinchpurse ethics and prudish morality. Twain was a rebel against American imperialism and the "manifest destiny" clapytrap which gripped our nation in his lifetime. Just as Poe's more fantastic literature is usually not reviewed in schools, so Twain's strongest condemnation's of American Policy in the Philippines after the Spanish War do not get taught in schools.
The War Prayer was an attack on unquestioning patriotism at the beginning of the war, and
To the Person Sitting in Darkness was a bitter diatribe against American policy after the end of that brief war, when American troops fought the Moro and Hukbalahap "insurgents." Twain also wrote a strong work condemning Mary Baker Eddy and Christian Science, after both his wife and daughter died in terrible agony, refusing all medical treatment because they had become devotees of that "religion."
Christian Science was withdrawn within weeks of its publication, and was out of print for more than 75 years. Christian Science may have made Protestant clergymen uncomfortable, but any attack on religion made them more uncomfortable. An excellent collection of Clemen's short work which takes a scalpel to society was published within the last ten years or so, and entitled
A Pen Warmed up in Hell--i highly recommend it.
Both Poe and Twain wrote in an American "idiom." That means that they used the speech patterns of the times and places in which they lived. Poe was much more the innovator literarily than was Clemens, but Clemens was not behind hand in exploring subjects "polite society" did not discuss, and his consistent scorn for religion and government means that most of his best work will never be taught in American schools. For his novels, i recommend
A Connecticutt Yankee in King Arthur's Court,
The Mysterious Stranger,
The Man Who Corrupted Hadleyburg and
The Tragedy of Puddin' Head Wilson.
Both these authors stand head and shoulders above all but a handful of American authors. The only problem with teaching about them in school is that the best of their work makes "normal" people uncomfortable, or is not understood by them, and so is not taught in school.