As early as the reign of Suleyman I (the Magnificent) in the 16th century there were ideas about the Ottomans not being Qurayshi and therefore not being eligible for the caliphate. The response from the Ottoman sadrazam (head-vizier) was that, simply, the Quraysh criterion was not a genuine one at all (Özcan, para. 3). The issue seems to have become more pressing during the late 19th century. During the reign of Abdulhamid II (a time of pan-Islamism and an unprecedented emphasis on the notion of Caliphate), many state officials and pro-Ottoman scholars proposed arguments to undermine the Quraysh criterion. Among their arguments were the approval of society (including religious scholars, soldiers, regular folk etc.), the will of God, the military and political power used for the good of Islam, and so on (Özcan, para. 9). Frankly, I do not know whether the Quraysh criterion was a genuine one or not. So I cannot tell you who was right in this debate; someone with more in-depth knowledge of the history of Islam might provide further information. What seems interesting is that Ottomans opposed the Quraysh criterion with arguments derived mostly (though not completely) from reality of their situation. They were the strongest Islamic state for a very long time; they were the ones repelling crusades in the Balkans and the Mediterranean, they were the ones protecting the Holy Lands, they were the ones organising and maintaining safe passage for Hajj. This, I believe, should also reflect at least part of the reason why the Ottoman Empire was recognised as the Caliphate. There were other entities claiming to be the Caliphate at the same time but none as widely recognised as the Ottomans and I reckon the Ottoman might must have had a fair share in bringing this about. This does not necessarily mean repression: it also means for some Muslims it might have been natural to simply accept the Ottoman Caliphate as natural. It is important to notice that contrary to common knowledge, the Ottoman sultans did not first use the title of ‘the Caliph’ after the defeat of the Mamluks at all: Murad II (father of Mehmed II the Conqueror) already used it, and from the reign of Murad I (1362-1389) the title was in use (not sure whether in ‘the caliph’ sense). It does seem that with no rivals of equal power in the Sunni Muslim world, the Caliphate claim became stronger. If there existed a completely independent Egypt that matched the Ottomans in might and claimed to be the Caliphate at the time of Abdulhamid II, for example, I imagine the latter would find it harder to convince the Muslims of the world. Especially if that Egypt happened to control the Holy Lands instead of the Ottomans. This is why, by the way, the Mamluk story is not wholly wrong. Ottomans called their sultans ‘the Caliph’ (among many other titles, of course) before the fall of the Mamluks, but this in turn (especially the control of the Holy Lands and becoming the undisputed major Sunni power) seems to have played a significant role in bolstering their reputation in the Islamic world.
Take the case of ‘Abd al-hamid al-Zahrawi, an Arab nationalist from modern-day Syria. al-Zahrawi published a book in 1901 on this issue in which we find at least adumbrations to the Quraysh criterion and the ineligibility of the Ottomans (Tarabein, p. 98). I can assure you that al-Zahrawi was not alone in arguing that the Ottomans were not Arabs/Qurayshi and therefore could not claim Caliphate (Ochsenwald, pp. 198-99). The Ottoman Caliphate served for other Arabs as the framework which would keep Turks and Arabs (among others) united under one state, of course. Again, as before, the fact that the Ottoman state represented the status-quo, that it was the strongest Muslim state, and it controlled the Holy Lands (and now it was under attack by ‘infidels’) must have been important factors for these people.