Here's an image that proves ancient schools did include MUSIC and the ARTS in their Curriculum. Note that the instrument shown has SEVEN strings and there also appears to be two cymbals on the wall.
For those who could afford classes or private tutors though in some polises basic literacy and some other courses were sponsored by the state the curriculum seems to have been basic arithmetic and some geometry, reading, music, dance, reciting Homer and some of the lyric poets, and then, more geometry and music and perhaps rhetoric and law.
Note that what we would call visual arts were taught by apprenticeship and there were no universities. The equivalent of tertiary educations was largely public.
Whether or not there were scholarships for the gifted is not known.
I suspect young men from families that could not afford tutors probably hung around the rare public free lecture taking notes and hoping to get the attention of teachers.
As for girls basic literacy seems to have been limited to the upper classes and wealthy and probably also extended to singing and dance training for public choirs and rituals.
so we are lucky to be able to have discussions about STEM versus STEAM and how much science and technology should be in the curriculum.
Support public schools not private academies or home schooling !
One thing we can certainly do better than the greeks is to NOT limit education to the wealthy!