An interesting point about this that a teacher recently made in a lecture I attended, is that so far no modern country has 'claimed' the Byzantine Empire as their national history.

Greece mostly identifies itself with the ancient polei and Turkey often portrays it's predecessor, The Ottoman Empire as a just victor over the corrupt, dying and backwards remnants of a once great empire.

Of course this goes for the Ancient Roman empire as well but that empire is usually seen as the universal ancestor of the West without one single nation having the biggest claim. Combine that with the historical rivalry between Constantinople and much of the Western World (The pope, Charlemagne, The Holy Roman Empire, Venice, numerous crusaders, Sicilian Normans, to name a few) and of course the Great Schism between what we now call the Catholic and Orthodox church, and you have a myriad of reasons why the Byzantine empire is usually no included in the notion of western historical identity.

So a combination of it's two arguable most likely succesors not claiming it's history as their own, and the long emnity between parts of western Europe and the Byzantines. I thought this was an interesting view on thr topic.

P.S.: Something something Edward Gibbon