Nope.
In the past soldiers fought for individuals, not countries. That is how the entire Imperial Roman Period worked and was such a cluster****. Once the emperors had lost their claim via bloodline it was up for grabs who could rally the most soldiers behind them.
Plenty of auxillary and foederati troops were non Roman. Plenty of legionaire were one generation removed from auxillaries who earned citizenship after their tour of duty aka were not technically Roman as well. While cultural assimilation was strong in the early empire, in the late empire it stopped working, partly because the Empire itself split into seperate cultural parts, in some cases actual political entities (GalloRoman empire existed for decades, Palmyrian Empire was a short lived, but strong Roman successor state)
Yeah, that is bollocks because the Catholics hated the Byzantines until the later were so weakened they decided to jump in. At that point Jerusalem had been under Muslim rules for a centuries and none of "Europe" gave a rat's ass.
Constantinople was Christian when the crusaders took it "back", destroyed the Roman Empire, established puppet states and left the rump region of East Rome in chaos.
If you speak of the later Habsburg Turkish Wars you have the Archdukes fighting over the corpse of Hungary with changing success, Ottoman's greatest ally? France, because of their squabbles with the Habsburgs in Spain, Italy and the Low Countries.
Wrong, plenty of Europeans fought with the Ottomans or are the Balkans suddenly non European?!? Moldovia, Bulgaria, Romania, Hungary, Serbia, Greeks, Bosnians... plenty of European vasalls who had significant impact on affairs in the Ottoman empire themselves.