I don't know where you lived, but this is the opposite of the situation in most places. Private schools attract people who will pay private school tuition because they offer higher standards.
In my limited experience with private schools, it was relatively easy to poach good teachers away from unionized jobs precisely because the school had freedom to offer them higher wages and better benefits. Many of them were also sick of dealing with the issues that come with unionized labor and union membership.
https://nces.ed.gov/programs/digest/d13/tables/dt13_211.10.a...
Sometimes I've seen them push back against bad management, but it is frightfully rare. Bullying, discrimination, favouritism, giving young/non-clique staff the worst jobs: all good.
I don't think public sector unions are very good stewards, partly because everyone is in the union. They push against govt objectives (control wages) and politics (push whatever rhetoric of the day is) and seek to gain public support (teaching/learning is essential for society etc). Unless you're a cog in the union machine, they won't be present except to prevent you being fired or make you work extra hours.
This became very evident during the pandemic: unions stuck with (1) better pay, (2) keep leave. While (1) is essential, the real problem was (3) death. But adjusting the school year, virtual teaching, classroom ventilation, support for students: nothing, except opposition. They wanted nothing to change. It was bonkers.