I personally agree with Kiyant back on the first page that Turkish should become a separate culture from the Levantine Group, and despite the many similarities with Greek, in the end, it shouldn't group with Greek either. Everyone is discussing the similarities between the Ottomans and Byzantium, while they seem to be ignoring the other Turkish nations that had more Arab, Nomadic, Persian influences, or even the few that held more or less to their Central Asian roots. There are many regional differences that aren't clearly divided or separate, but the comparing of all Turkish peoples in Anatolia to Greeks is... less than ok. Please correct me, as I am probably wrong, but Dulkadir, Aq Qoyunlu, Ramazan, Karaman (and Qara Qoyunlu, if their primary culture changed to Turkish) have a much larger disparity than the OE, yet they are also Turkish (for all intents and purposes). Is there as much Greek influence in Eastern Anatolia than the West? Do these countries really hold much tie to the Greeks conquered in the previous centuries? Splitting the Turkish in two or three (to me) seems to make the most sense. Also, if using religion (Orthodox means Greek, Sunni means Turk), uniting a Greco-Turkish group makes even less sense, as then the associated "cultures" are considered to be completely divided, as Sunni converts quickly adopted language, customs and frame of mind to 'fit in'.
Now whether Azerbaijani/Azeri is also considered part of the Turkish group is not as much of a complaint, as they are Heavily influenced by Persian culture. But it could go either way.
You have some valid arguments based on actual history and culture, but fail on "gameplay reasons". That is why Azerbaijani and Turkish are in different culture groups despite speaking variations of the same language.
The Sheeps' interest is to the east. When you look at the updated culture and politic maps below you will also see that they already sit on Armenian and Kurdish populated lands (btw Karaman also has famous sheep, which reveal its tricks later, but that is not our subject atm

)
A Byzantine Culture group from Albania to Armenia would be similar to the Carpathian Group, making a
unit sharing the same core for centuries despite the cultural disparity. Turkish culture works as a tool that connects Armenian to Greek by various layers of influence (we have delicious food aside from the "kebabs", combining Greek and Armenian kitchen). As mentioned earlier, nomads generally don't have a heavy culture to impose (but influence some), and rather adapt to the locals when they settle.
The quick and perfect conversion you assume is not very realistic. Even in this global world, if I migrate to England and try to be as English as I can be, I would definitely fail and come up with a "unique" version of Englishman for myself. It would at least take one generation to truely adapt, and that is when there is already a strong, established culture present. All those Greek conversions by the time created a heterodox way of Islam in the region, very much different than the Arabic world and made up a unique culture. Turks themselves were new to the religion, some of them even still carrying on their shamanist ways.
Even if Ottomans hadn't converted them, Ionians were not the same as mainland Greeks anyway. "Beware of the Greeks bearing gifts" is a famous Troyan proverd

. So it is OK to have some disparity between the Dulkadiris and the Ottomans despite being in the same culture
group I guess
.
The pink parts are the Byzantine Empire in 1020:
This is Mehmed II.'s Empire in 1481:
(that heel of Italy -Otronto- was also under control of Ottomans for a while btw. Mehmed assumed himself as the Roman Emperor).
This is the core of the Ottoman Empire, and making Otto the cultural union for Albanians, Greeks, Anatolians and Armenians would make more sense imho. Neither of these cultures fit well into the culture group they are currently in, yet lived in the same empire without major conflicts for around 300 years (approx: 1450-1750). All the southeast stuff for the Ottos come 70 years after the game start. Turks to the East were not kin, they were enemy.
If you combine the above map with the Abbasid Caliphate map...
...you will get almost perfectly matching borders.