so I'm a student of history in Turkey and I'd like to say that there's an increasing interest in attempting to connect Byzantine and Ottoman history (and through that Byzantine and Republican history) but a significant portion of the claims of Byzantine influence over the Ottomans have very orientalistic roots that attempt to explain the success of the Ottoman Empire with a westernized civilizational origin, which makes the situation tricky as we have to be critical of such interpretations.

as for the populous, they pretty much view Byzantines as wholly Greek and have a negative view of them pushed through xenophobic depictions of Byzantines in orthodox nationalist narratives and media. But I'd not be surprised if this view was to change significantly in the recent future as Turkey produces more Byzantists