Does the South Dakota Legislature have the duty to establish an educational system?

Yes.  Article VIII, Section 1 of the South Dakota Constitution emphasizes the importance of a “general and uniform system of public schools” and places the duty to establish the system on the Legislature:

The stability of a republican form of government depending on the morality and intelligence of the people, it shall be the duty of the Legislature to establish and maintain a general and uniform system of public schools wherein tuition shall be without charge, and equally open to all; and to adopt all suitable means to secure to the people the advantages and opportunities of education.