It's actually worse than that. The NYCBOE is not actually a state agency - it's an "administrative body" that's comprised of people selected by the two main political parties, and staffed by their family members. After they are appointed, they aren't under the oversight of any elected official.[0]

You're correct that the buck stops with Cuomo, though - changing this system would require him authorizing legislation at the state level. Due to the way New York's legislative process works, it's impossible for legislation to even reach a floor vote in the Senate without the governor's approval; the governor can essentially veto a bill behind closed doors by refusing to let it have a floor vote.

[0] https://www.nytimes.com/2020/10/26/nyregion/nyc-voting-elect...