It serves the same political purpose that it did half a thousand years ago. It serves the same purpose that it did a thousand years ago.
The Byzantines are the surviving part of the Roman Empire. What the western kingdoms think of them is the opinion of barbarians*. What the Papacy thinks is hardly decisive either.
They have virtually no meaningful connection with the pre-Roman Greeks, and I'm including the Byzantine literary fetish for that style here. Identifying as Greek over Roman would be throwing out the state's identity to replace it with something it doesn't even have any real ties to.
* Anna Comnena's word when describing the Franks, at least in English translation.
I have "Alexias" both in Greek(hense the title) and the English translation.
The term is correct and it reflects the way the Greek society of Constantinople saw 'these strange Franks' "who wore animal skins,were unwashed and called themselves nobles"(witness anonymous) nevertheless
the opinion prevelent at that time didn't stop the goverment and merchants to deal with these people and the Byzantines knew very well how the pope influenced the public and ruling class in the west at that time and that basically,these barbarians were hostile to the 'heretic Greeks'.
The opinion you site about the Greeks associates with the opinion of the church that equates the term 'Greek' with the term 'pagan' and those days church was a very powerful organization that even emperos didn't dare cross openly;naturally the common men used the term out of fear of the church mostly, but an inhabitant of Athens new he was a Hellene,a Peloponnesian knew he was a Hellene etc,also see the lords of Romania(peloponnese)calling the Greeks with the derisive term 'Griffons' because they knew they thought themselves Greeks,if they thought themselves Romans they would not be offended by the epithet 'Griffons' since they would feel that it didn't apply to them...
The hold of the church over such matters was wearing off in the late empire since the intelligencia headed my Mihael Psellos knew and called themselves Greeks,and referred quite derisively to those poor ignorant Romans who had turned barbarians, meaning the whole of Italy,the Pope included.There are documents in literature that are in various libraries and monasteries that write clearly all these events...but of course you are rather persistent aren't you?