The answer is no, there is no requirement for a DoEd Secretary to support public schools.
This question misunderstands the reasons the Department of Education exists, it's core mission, and the role of the federal government in education generally. The mission is
Congress established the U.S. Department of Education (ED) on May 4, 1980, in the Department of Education Organization Act (Public Law 96-88 of October 1979). Under this law, ED's mission is to:
Strengthen the Federal commitment to assuring access to equal educational opportunity for every individual;
Supplement and complement the efforts of states, the local school systems and other instrumentalities of the states, the private sector, public and private nonprofit educational research institutions, community-based organizations, parents, and students to improve the quality of education;
Encourage the increased involvement of the public, parents, and students in Federal education programs;
Promote improvements in the quality and usefulness of education through Federally supported research, evaluation, and sharing of information;
Improve the coordination of Federal education programs;
Improve the management of Federal education activities; and
Increase the accountability of Federal education programs to the President, the Congress, and the public.
If anything, given the repeat failure of some public schools to deliver anything like a quality education, I would think it incumbent on the Secretary to have an open mind about alternative ways to deliver that education and that would be consistent with the quality mandate.
The Secretary has come under scrutiny for her lack of candor on her required financial disclosure forms and related testimony. That could potentially lead to a conflict of interest down the road. However, it's her massive financial holdings that would so activate the conflict, not her position on public schools.