Comment by spydum
I can attest to this exact same scenario with my children and their schools. I observed both types: the parents who immediately entered the school combative towards administration (not looking to collaborate on a solution, just shouting loudly to "fix it"), and parents who spent time engaging with administration towards a description of the problem and ideas for resolving them.
That being said, there are and were definitely limits to what public schools can do. They are resource strapped, procedurally constrained, often fighting their own political bureaucratic battles within the school districts, and even within the academic departments.
We ended up leaving the public school for those reasons, and could not be happier.
My observation was: public schools have become much like enterprises, and private schools tend to be more like startups. The public school has so much inertia and tends to have "guardrails" and policies to keep even bad administrations functioning, but at the cost of exceptionalism and performance. Private schools have less of this, and more direct accountability.
You absolutely can have a private school that doesn't educate better than a public school, but I'd argue at least one of two things happen: 1) the school fails to attract student, and closes.. or 2) the school focus shifts away from education to other priorities (e.g., social status, culture, or sports), relegating academics to a secondary role.
I taught at a private school for several years, and what I saw there convinced me that they are up against many of the same things that public schools are except they need a marketing budget and all the parents have financial resources (although few had the resource of time).
I guess I think it's hard to make real generalities about public v private schools, and good data is hard to come by since stats can be juiced in various ways for various reasons.