This is not history but ahadeeth and scholarly interpretation of them, some of which from your scholars or scholars from the first era who we all look up to as reliable scholars.Sorry my friend but this is our history and we know it.
You can dispute about Ottoman Empire, but don't say the Prophet said Caliph has to be from Quryash since i can't find anything like that in ahadeeth.
As for legitimacy of Caliphates, I don't know what we mean by 'legitimacy' since it is just a form of government. Whoever forced their way could take it over or declare one by theirselves. As we know after the first four Caliphs the Caliphate system was exploited and people just wanted power from there on. Many of those being Arabs people who were not sincere about God or Islam.
So 'legitimate' Caliphate doesn't make sense since anyone with power could take it over or do a coup and so on. This is why the Prophet(SAW) made distinction between regular Caliphate(just regular empire) or Caliphate on the path of prophethood, which is a Caliphate led by a very righteous and just person which God approves of to rule over people. For me personally I don't care about empires but doesn't mean I hate them or the people that formed them. But only thing stands out to me is someone who is obedient and righteous ruling Muslims which will make a difference and we will see and understand why Allah(SWT) told us to observe the right path and how it will brings his blessings to us if we do.
Anyway I think you should keep your dispute with Turkey/Erdogan aside for this case because it is not related to disputes you guys have together. Some Muslims support Turks here because they feel Syrians have been oppressed, doesn't mean they see Erdogan as amazing Muslim or his party. And we don't need to talk about Ottoman Empire.
Let's stick to topic since it is very crucial period in Syrian conflict and may develop into bigger war that will affect whole region.