Wages are only part of the function of a union. An arguably more important part is preventing arbitrary and capricious management decisions, including retribution against employees for expressing grievances. A big part of what unions do day-in, day-out, is dispute resolution. A unionized meat factory isn't going to pay much better wages, if at all, than a non-unionized factory; but it would be significantly safer.

The thing is, the courts often grant Due Process protections to government employees on the theory that a government employer taking actions against an employee isn't too far removed from the government taking actions against any random citizen. At a minimum, the government almost always must follow some sort of reviewable investigatory procedure. Compared to at-will employment, the burden (however de minimis in an absolute sense) is much greater. The need for checks against malicious government managers is significantly reduced.

The upshot is that unions aren't as necessary in the case of government employees. The downshot is that even if you prevented unionization, the government would still have problems firing problematic employees. The issue is never that they can't be fired (union or not), but whether it's worth the hassle, which has a cost in terms of time & money.

> A unionized meat factory isn't going to pay much better wages, if at all, than a non-unionized factory; but it would be significantly safer.

Unionized workers make more than their non-union peers, and have significantly better benefits.

You can see a huge difference in pay and benefits with union and non-union construction workers and welders. One group is paid a living wage with benefits, and has options for retirement, and the other group makes barely more than minimum wage, has poor benefits if they're even classified as employees and not contractors, and they're on their own when it comes to retirement.

Similarly, white collar unionized workers make more than their non-union counterparts, too. They have higher pay, better benefits, and more paid time off.

"The state wants to minimize budget"

This is a joke statement right? There is no state in the history of the world that attempts to do this.

What nonsense, firstly OP clearly is specifically referring to the budget outlay for education, and even if taken more broadly I can assure you throughout the world there are plenty of governments whose spending is constrained by current income and who very much do aim to minimise expenditure for a given outcome (predictably often with the effect of greatly worsening outcomes).

Maybe you'd prefer it reframed as: elected officials want to minimize spending on line-items they don't care about in order to spend on things they do care about.

What are you talking about? Education budgets are almost comically prone to budget cuts, it's one of the most contentious facets of government spending (at least in the U.S.)

Maybe not in the states you’ve lived in, but this was very much the hypothesis behind Sam Brownback’s experiment in Kansas. Beliefs in a strong version of the crowding out effect result in states trying to cut their budgets.