"Ethnic" identity in the Middle East in the pre-modern and into the modern era often had much more to do with language and religion than it did with any of our contemporary genetic conceptions. (More on that in a second.)

So for example the often fully ethnically Circassian Mamluks of Egypt in the Middle Ages were often identified as "Turks" by the local populace because they spoke Turkish.

As for the Ottoman Turkish Sultans, the mother of the Sultan was known as the Valide Sultan. You can see a list of them here.

As you note, genetically the Ottoman Sultans would have been much more closely related to, say, the Circassians and other peoples from the Caucuses that made up much of their matrilineal line than they would of any Turkic people of Central Asia.

The thing is: this is largely true of the population of Turkey as well, as DNA testing is increasingly making very obvious. This article, for example, might be of interest.

The fact is that the existing populations of Anatolia didn't disappear just because the Ottomans showed up. They often converted, learned Turkish, intermarried, and became "Turks".