[DuMOHsmol]

Slavic languages originally used the very unique Glagolitic script, developed by saints Cyril and Methodius. However, their disciples later created an alternative Greek-based script, which eventually prevailed.

My guess is that the educated people of the time were very familiar with Greek, so it was easier for them to work with Greek letters rather than the newly invented ones. It probably was the same for Latin-based scripts as well.

[mananaysiempre]

Worth remembering that all of the people you mention were Bulgarian Eastern Orthodox monks and missionaries, so for them Greek was both the international vernacular and geographically close.

[userulluipeste]

Cyril and Methodius were in fact Greek Christians, baptized as Constantine and Michael. They only later adopted monk names that they are ultimately remembered as. Now, considering that the Byzantines (or "Eastern Romans") were in conflict with Slavs since long before the brothers were born, it's unlikely for their family to be of Slavic background and at the same time for their father to be in a prominent military position (droungarios).

The script that came to be known as Cyrilic was developed by Cyril and Methodius' followers, after they were exiled from Great Moravia (by the bishop that replaced Methodius), so it's safe to assume that those in that group were not Bulgarians either.