[matthewgard1]

between the microschool and ESA movements going on right now, there will likely be many publicly-funded & worker-owned schools popping up over the next few years (amongst other much stranger hybrids).

In many jurisdictions, a teacher co-op can already obtain a charter to have an "open to the public, funded by the public, and accountable to the public" charter school. If we're going to be pedantic I think that would fit your funded+governed definition. owned+operatred might be closer to what you are gesturing at though, or perhaps local democratic oversight? Regardless, the old public/parochial types of school categorization is not nearly nuanced enough to be particularly useful for where things are already, let alone soon headed.