In school before 16, our only exposure to older forms of English was Shakespeare, which as you point out, is early modern English. However, since I elected to study English at A-level (16-18) I studied Chaucer (Middle English) and a little bit of Saxon authors. To say that I studied OE would be false though.
To a modern English speaker, generally anything written since 1800 is perfectly readable. EME is readable in the most part but certain turns of phrase and words need to be explained; books of Shakespeare's works often come with footnotes explaining strange grammar or orthography. Middle English - well you can get maybe 40-50% of that, more if you know French as there are lots of French words used. I have a book of Chaucer's poetry and I often have to look in the glossary as it gets incomprehensible:
"O foule lust of luxurie! lo thyn ende!
Nat only that thou feyntest mannes minde,
But verraily thou wolt his body shende,
Th'ende of thy werk or of thy lustes blinde
Is compleyning, how many-oon may men find
That noght for werk som-tyme, but for th'entente
To doon this sinne, ben outher slyne or shente!"
[From The Man of Law's Tale in the Canterbury Tales]
As for Old English, pre-1066, that is incomprehensible to a modern English speaker even if you speak German/Scandinavian languages, from which most of the vocabulary came. To use a passage from the introduction of the Anglo-Saxon Chronicle (around 900AD), from two whole paragraphs I can pick out meaning from maybe ten or fifteen words; "his sunu" (his son), Westsaxena (West Saxony), cyningas þe (King of) - and that's because I've studied it before! In short Old English is not useful to us at all because it is a completely different language to us. English has changed a lot more than Chinese has and so we tend to stick to learning the more modern forms of the language.