In the WW1 era when this phase was used to describe the Ottomans, were they really considered a part of Europe? Would North Africa be considered similarly? Live been trying to reconcile what, as a generic American, I see as a major change to now when Turkey ( and North Africa) seems extremely "other" from Europe.
I think part of the issue is just seeing the Ottoman borders in 1914 and not at their height, when they were very much a geopolitical player in Europe if not a completely cultural member, while of course Europe itself was never truly uniform, but to say the least was never a member of Christendom.
In 1800 even the Empire still ruled much of the Balkans See Map, and while a shrinkage from their height, of being at the gates of Vienna, it was that last 100 years that would involve much of the slipping away of that territory. Even in 1912 with the First and Second Balkan Wars the Ottomans would be seeking to better their position and dealing with breakaway states like Albania, Bulgaria, and Greece(one of the earliest states to agitate and fight their way free in the 1820's). While the sick man title might also be a bit uncharitable and simple slander from the likes of Britain who bounced between seeking to promote the power of Christian majority territories of the Empire, but also in using the Ottomans as ally's against Russian ambitions which led to their involvement in the Crimean War.