It is hard to exactly say why the Ottomans adopted the devşirme system, as our records around its founding are quite slim. Due to this, the founding of the system has been attributed to various early Ottoman amirs and attributed to various reasons.
While the origins are still up for debate, some have argued that the creation of this system was a way for the sultan to consolidate power. Previously, the Ottoman state had relied on a series of semi-independent leaders, who contributed soldiers to the Ottoman army. Under the devşirme system, the Ottoman state directly controlled a small, loyal army and an array of local administrators.
Doubtless, this did not entirely consolidate power, as the janissaries never made up the entirety of the Ottoman army. In fact, the number of janissaries was quite modest in the beginning, During one of our first records of the system, Isidore Glabas' 1395 sermon denouncing the practice^(1), the janissaries were estimated to only be 2,000 soldiers strong.^(2)
Comparatively, the total Ottoman force at the Battle of Kosovo, in 1389, is often estimated around 20,000 to 30,000 soldiers. Therefore, the sultan's personal army would only make up about 1/10 of the entire army.
Regarding the ethnicity of those enslaved by this system, we have records of not only Greeks (including Pontic Greeks), but also Albanians, Serbians, Bosnians, Bulgarians, Armenians. So, as you can see, the devşirme system was incredibly widespread throughout the Ottoman state and affected more than just Greek communities.
Notably, the major affected Armenian and Pontic Greek populations resided in Anatolia. Therefore the system was not restrained to the Balkans.
Even though this is a culturally diverse list of affected peoples, the common factor was that Ottomans drew slaves from communities that were Christian. But there is one exception to this. While many Bosnians had converted to Islam follow the Ottoman conquest of the region, Muslim Bosnians were still subject to the *devşirme.*^(3)
A second special case were the Christians of Wallachia and Moldavia. Both of these states became Ottoman vassals as the Ottomans expanded further into Europe. While they were Christian, they were given a special exemption from the devşirme. However, Wallachia had previously paid a tribute of 500 children per year to the Ottoman state under Vlad II Dracul.^(4)
Bibliography:
^(1) Isidore Glabas and the Turkish Devshirme (Vryonis), see page 437. This is a bit of a dated work, but contains a great translation of the sermon.
^(2) Fighting for a Living : A Comparative Study of Military Labour 1500-2000 (Veinstein), see page 121, see section "Figures".
^(3) Ottoman Institutions, Devshirme: 1250 to 1920: Middle East (Ali), in Cultural Sociology of the Middle East, Asia, & Africa: An Encyclopedia.
^(4) Dracula, Prince of Many Faces (Florescu).