It’s pretty smart to make the things that benefit the weak and hurt the powerful illegal.

That argument doesn't hold water when were talking about public sector employees. You don't become a public worker to get rich. Thusly being a public worker, the government can do as they please. And unlike a private company, the government can break the law.

Unless I'm missing something obvious, striking is the only leverage a union has. If striking is illegal, what's their leverage?

The ATC union is a monopoly, which is just as bad as a business that has a monopoly. They can hold the entire aviation system in the US hostage. This is why it is illegal for them to strike.

They knew when they took the job that striking was illegal. Reagan reminded them of that, and said he'd fire them if they broke the law. They did anyway, and Reagan wasn't bluffing. Reagan could have also prosecuted them, but (wisely) chose not to.

Firemen and police officers are also usually not allowed to strike. For the sake of continuation of society.