This is going to sound dismissively cynical, but it all comes down to politics. The public school system is accountable to way more entities than the Catholic school is. Teachers Unions wield a tremendous amount of power and influence in local and state government. Actually taking on such a political machine would require an incredible amount of activism - bordering on replacing the majority of elected positions in your district. This is especially difficult because most of the lower positions are seats that the average person never thinks about or votes in. But, the unions make sure their members know about these positions and endeavor to get out the vote.

The Wisconsin state government managed to pass Act 10 in 2011 which greatly reduced the power of teachers' unions.

https://en.wikipedia.org/wiki/2011_Wisconsin_Act_10#Effect_o...

it’s been 10 years, do we know how that’s affected the quality of education there?

Very very badly.

"the law reduced average test scores on the state’s standardized exam by approximately 20% of a standard deviation": https://www.sciencedirect.com/science/article/pii/S027277571...

"8 percent reduction in total compensation" https://www.cato.org/sites/cato.org/files/pubs/pdf/rb71.pdf

Not only does this hold up in general, it holds up district by district. Districts that still had collective bargaining had better scores than one that didn't (even after controlling for confounds, it was the union status that made the difference).

https://journals.sagepub.com/doi/abs/10.1177/0160449X1988337...

Not only that, teacher shortages are worse, and they're much worse in rural areas.

https://search.proquest.com/openview/5ae518d77f9e44551206208...

So the outcome is: Teachers make less money, there are fewer teachers to be had, and students are scoring much worse.

A win-win-win for conservatives!

The public school board is an elected body; you can show up at your city or town or village's election and pick the members you like. In practice, few voters are invested parents, and nobody pays attention to school-board candidates most of the time. (If you follow the news, Loudoun County, VA is a notable recent example of how things can go wrong.) So many public school districts are stuck with the same corruption that occurs in any public position.

The Catholic schools' policy on things like this, at least near me, is dictated mostly by the diocese. The state can withhold funding for things like special-ed programs or meals if they won't play ball, but they get more autonomy as long as they aren't violating civil-rights law.

> In practice, few voters are invested parents, and nobody pays attention to school-board candidates most of the time.

The teachers' union does. Not most of the time either. All the time.

Because private school parents can pull their kid out and take their tuition elsewhere. They actually have options and can make good on bargaining threats to walk away.

Additionally, you don't have public sector or really any teachers unions to deal with at private schools.

There are other factors that contribute, but aren't nearly as clear cut as those two.