Why would people hired by Mark Zuckerberg to fix the school system be any better

Because he, presumably, would select them by different criteria and because he can fire them much much easier than a politician can fire a union-entrenched educrat.

while Zuckerberg's employees only need to please their employer

This is precisely what creates an opportunity for him to be effective.

As opposed to the outcomes of charity-funded school systems?

So do tell, what do you think is the problem?