I've read a lot of those before I was 18, and some of them were dire beyond all description.
'Thus spake Zarathustra' for one. And none in languages other than English in all its historical varients. Plus I had to reread a number, because although I read them the first time round and saw some performed, I hadn't fully understood the text.
But yes, it's an American list for showing off, and I'd be fascinated to know how many of the adults recommending it have read all the texts on it and understood them.
Bet it's not many. 
As for feeling thick if you haven't read any of them, why? Reading books is one skill amongst many.
Ninth grade, BC 5000-400 AD
The Bible: Genesis, Job
Epic of Gilgamesh (c. 2500 BC)
The Iliad and The Odyssey, Homer (c. 850 BC)
History of the Persian Wars by Herodotus (485-424 BC)
Sophocles, Oedipus Rex (c. 440 BC)
Medea, Euripides (c. 431 BC)
The Frogs, Aristophanes (405 BC)
Republic, Symposium, Plato (c. 387 BC)
On Poetics, Ethics, Aristotle (384-322 BC)
The Bible: The Book of Daniel (c. 165 BC)
The Aeneid by Virgil (c. 30 BC)
Metamorphoses by Ovid (c. 5)
The Bible: Paul, 1 & 2 Letters to the Corinthians (c. 58 AD)
The Wars of the Jews by Josephus (c. 68)
The Annals of Tacitus (c. 117)
Tenth grade, 400-1600
Augustine, Confessions and City of God, Book 8 (c. 411)
The Koran (selections) (c. 650)
Beowulf (c. 1000)
The Mabinogion (c. 1050)
Aquinas: Selected Writings (ed. Robert Goodwin) (c. 1273)
The Inferno, Dante (1320)
Gawain and the Green Knight (c. 1400)
Chaucer, The Canterbury Tales (selections)(c. 1400)
Malory, Le Morte d’Arthur (selections)(c. 1470)
The Prince by Machiavelli (1513)
Utopia by Thomas More (1516)
Faustus, Marlowe (1588)
The Faerie Queene, Spenser (1590)
Julius Caesar (1599), Hamlet (1600), or other plays, Shakespeare
Eleventh grade, 1600-1850
Cervantes, Don Quixote (abridged)(1605)
Divine Meditations, John Donne (c. 1635)
Paradise Lost (selections), Milton (1664)
Pilgrim’s Progress by John Bunyan (1678)
Gulliver’s Travels, Swift (1726)
Songs of Innocence and Experience, Blake (1789)
“The Rights of Man,” Paine (1792)
Lyrical Ballads, Wordsworth and Coleridge (1798)
Pride and Prejudice, Austen (1813)
Frankenstein, Mary Shelley (1818)
“Ode to a Nightingale” and other poems of Keats (1820s)
The Last of the Mohicans, Cooper (1826)
“The Lady of Shalott” and other poems of Tennyson (1832)
“The Fall of the House of Usher” and other stories of Poe (1839)
Jane Eyre, Charlotte Bronte (1847)
Moby Dick, Melville (1851)
Twelfth grade, 1850-present day
Communist Manifesto, Marx and Engles (1848)
Uncle Tom’s Cabin, Stowe (1852)
Leaves of Grass, Walt Whitman (1855)
Crime and Punishment, Dostoyevsky (1856)
On the Origin of Species, Darwin (1859)
Great Expectations, Dickens (1861)
War and Peace, Tolstoy (1864)
The Return of the Native, Hardy (1878)
Thus Spoke Zarathustra, Nietzsche (1883)
Huckleberry Finn, Twain (1884)
Selected Poems, W. B. Yeats (1895)
The Interpretation of Dreams, Freud (1900)
“The Innocence of Father Brown,” Chesterton (1911)
Selected Poems, Wilfrid Owen (1918)
“A Poem with Notes and Grace Notes,” Frost (Pulitzer, 1924)
“The Trial,” Kafka (1925)
“Murder in the Cathedral,” T. S. Eliot (1935)
The Grapes of Wrath, Steinbeck (1939)
Mein Kampf, Hitler (1939)
Animal Farm, Orwell (1945)
The Diary of Anne Frank, Anne Frank (1947)
Invisible Man, Ellison (1952)
“The Crucible,” Miller (1953)
“A Man for All Seasons, Bolt (1962)
“Rosencrantz and Guildenstern are Dead,” Stoppard (1967)
“The Gulag Archipelago,” Solezhenitsyn (1974)