I’m somewhat confused by several answers talking about the present day status of Istanbul, or Golden Dawn’s vision of retaking the City.
Greeks may continue to refer to Istanbul as Constantinople (except for the Rum that actual live there), but most of them do know the difference between the Byzantine city of yore and the modern Turkish city.
And the dividing line between the two, the 1453 Fall, is still a defining event in how Greeks see their identity. It is a disproportionate reaction to what actually happened historically: the real damage was done in 1204, and the city state of 1453 was not worth salvaging. If anything, it is an insult to the thousand years that preceded it: what Greeks have come to care most about the Byzantine Empire is that the Turks conquered it. And focusing on what you have lost is not how you go about standing up on your feet again.
A lot of the focus on 1453 has been driven by nationalist education. Even more of it has been driven by the need for a creation myth for the hostility between Greeks and Turks, which has deep roots.
Yet myths do matter. Almost as much as history does. It remains a signpost, and it remains a Shrine of folk memory.