I ask this because for all the narratives of Ottoman conquest, rule and brutality, the Balkans remain predominantly with their European cultural identities. The peoples of the Balkans don't identify as Turkish or Arabic, they don't speak said languages either and Islam is a very insignificant minority overall
The reason for this (or part of the reason for this) is actually within your question.
Romantic-era ultra-nationalists who would evolve into far-right monarchists and fascists in the 20th century.
The Ottoman political, economic, and cultural impact on the Balkans was incredibly sweeping. You can find a short overview of Turkish influence (through the transmission of not only words but even grammatical particles and suffixes) on the Serbian language
in this article, which also discusses the bilingualism that defined many areas in rural and urban Serbia with mixed Serbian-Turkish populations. Although the article does go on to talk about attempts to edit the Turkish side out of literary Serbo-Croatian these efforts honestly don't have that much popular impact. There's Serb-nationalist songs from the Yugoslav Wars where they sing about annihilating Europe's Muslims and
still use Turkish loans in the lyrics. If we want to talk about religion, Islam was and is a lot more influential than you think. Rather than being an insignificant minority, Balkan Muslims still number in the millions, and include such politically significant groups as the Bosniaks (who also remain a significant minority in Serbia's Sandzak region), the Albanians (who can be spotted in Albania, Macedonia, and Kosovo sporting such Turkish names as Enver and Talaat) and over
1.5 million ethnic Turks in Bulgaria, 340k of whom
departed the country after the end of communism. There's almost no field of Balkan life which remained unaffected by Turkish rule.
Now, how does such influence exert itself? Through conquest, and not vassalage.
This graduate thesis (I'd like to find a more respectable source but I lost my reference doc a while back) is focused on the power structure within the 1700s Ottoman Empire, but Chapter 2 (page 40 of the PDF) gives a rundown on Ottoman administration in earlier centuries. The common trait of both time period, however, is that the Ottomans did not use unaltered Balkan administrative systems as intermediaries between Istanbul and the village, but instead co-opted/destroyed those systems to create new intermediary institutions with more Ottoman influence.
As this book also explains, the power of the Ottoman state and its functionaries over the Balkans was closely tied to the ability to hold and redistribute land, with such institutions as the
timar giving a legal basis for the process by which Ottoman functionaries profited from Balkan land and labor. Of course, when I say "Ottoman functionary" I don't necessarily mean a Turk. The existence of Slavic and Greek nobles within the Ottoman structure is well-attested and actually goes way back to the very founding of the Ottoman Empire, when Osman and Orhan (then only the Beys of a small fiefdom, now regarded as the first and second Sultans of the Empire) conducted marriage alliances with Byzantine notables to secure land and intervene in Byzantine politics. But while a Balkan noble living within the Ottoman provinces (these go through various names-- at first they are beylerbeyliks, then eyalets, then sancaks) could be a high-ranking official or even a Grand Vizier (
@Koprulu Mustafa Pasha), they could never be a king or a prince. In the pre-1800s Ottoman society, political sovereignty rested with the Padishah in Istanbul. In fact, the rise of Balkan "viceroys" or "vassals" in the 1800s (The Principalities of Serbia, Montenegro, Bulgaria, and Eastern Rumelia) are associated with the rapid
decay of Ottoman authority in the Balkans in the face of internal and external stresses (Atalomos also mentions this on his post in this thread's page 1). Even in the face of that decay, however, it's worth noting how influential
Ottoman land-use techniques remained in "independent" Serbia under Milan Obrenovic.
I guess that the closest the Ottoman Empire at its prime (pre-1800s) gets to using "vassalage" (ruling through cooperation with locally-based and autonomous systems) as an administrative technique is in their use of ecclesiastical control. Under the millet system, most of the previously autocephalous Orthodox chrcuhes were steadily folded into the
Patriarchate of Constantinople, the Greek-dominated proxy-church through which the Ottomans kept an eye on the Slavic/Vlach Orthodox (for more info on Ottoman-facilitated Greek influence in the Balkans,
check out this). As with the rise of autonomous viceregal territorial units, the re-creation of autocephalous Slavic Orthodox churches in Serbia and Bulgaria is associated with the 1800s decay of Ottoman authority over the Balkans and the machinations of foreign powers.
But for all this, you're right in saying that the Balkan states are now "European" and majority-Christian. That's how they are
today.
The reason for that is massive ethnic cleansing, in which millions of Balkan Muslims (ethnic Turks, converted Slavs, and everything in between) were killed or forced to flee. Today, they live in Turkey as the
Muhacir ("immigrant," "refugee") people. The first wave of Muhacir migration (which also included Circassians from the Caucasus) may have been up to
10 million people. Many of the Muhacirs are Pomaks, a term which refers to Bulgarian Muslims specifically. In other words, the creation of ethnically homogenous Christian nations in the Balkans wasn't a simple process of removing a superficial layer of Ottoman influence. It was a deeply traumatic process by which societies were forcibly remade, more comparable to tearing out an organ than picking off a scab. The blame for the initiation of this process lies more with local actors than with the foreign powers that egged them on-- the national elites of the new Balkan states successfully created ideals of "Serbness" and "Greekness," and made adherence to those ideals essential for political and social advancement. But the thing about this process is that although it tries to project a superficial image that the Ottoman conquest wasn't that impactful and never dulled the edges of Serb/Greek/Macedonian/Bulgarian national unity, dig a little deeper and you'll find that that image is a myth, and one that was destructive for nearly everyone involved regardless of religion.