Keep in mind that Constantinople was quite depopulated by the time it was conquered, not the capital of an empire. Inside the ancient walls, the city was a hollow collection of villages and gardens. The last emperor, Constantine XI, commissioned his assistant to conduct a census of all the able-bodied men in the city (EDIT: just prior to the siege) and he came up with slightly under 5,000 native Greek men and another 2,000 foreign mercenaries/volunteers, which was so disheartening he tried to suppress the knowledge. There were probably about 50,000 people total living in the city.

That said, you're right that the Turks were generally tolerant. Mehmed II was keen on revitalizing Constantinople as his imperial capital and allowed Greek refugees to return. A whole class of Greek administrators arose called Phanariots, named for the Phanar quarter in Constantinople where they lived. They provided the Ottomans with Patriarchs, governors, and diplomats.

And if the Ottomans didn't clear them out, how did it end up majority Muslim anyway?

At the turn of the 20th century, Constantinople was still a multiethnic city with large Greek, Armenian, and Jewish communities along with slight Muslim majority. After World War I, Greece invaded the collapsing Ottoman Empire intent on conquering areas with significant Greek populations, but they were defeated by Kemal Ataturk's nationalist forces. This led to population exchanges where Greece expelled the Turks remaining within its borders and vice versa. Greeks remaining in Turkey were increasingly discriminated against culminating in the Istanbul Pogrom of 1955, which led to most of the remaining Greeks to immigrate. Now there are only a couple thousand left, along with the Patriarchate.