The Greeks do have some connections to their Byzantine/Roman heritage mostly in regards to names of offices, people, locations, etc. The reasoning for the strong push to orient Modern Greece towards Ancient Greece lies with the Great Powers and the Philhellenes who really disliked the Byzantine Empire. They considered the Byzantine Empire to be this decadent, oppressive, corrupt entity, while at the same time praising Ancient Greece as this sophisticated and cultured society. As the Powers held great power and infleucne over Greece both IOTL and ITTL there will be a strong Classical Greece flavor although probably less so than OTL given the lack of Otto of Bavaria, who was a major supporter of Ancient Greece over Byzantine Greece.
Another thing that doesn't help is the fact that the Ottomans adopted a lot of the Byzantine Empire's institutions and bureaucracy for themselves when they took over. The Ottomans also hold a lot of the core territory of the Byzantine Empire, namely Constantinople, Thrace, and Asia Minor. Should the Greeks ever secure that then a bit of a Byzantine revival could occur in Greek society.
To tell the truth I'm at somewhat of a loss on what exactly this is supposed to be. The average Greek of the time or later is not feeling much of a conflict between his ancient and Byzantine identity. And Constantine Paparrigopoulos https://en.wikipedia.org/wiki/Constantine_Paparrigopoulos is just around the corner.