What Reagan did in firing the air traffic controllers was legal - yes technically. The fact that he violated terms of the FLRA during the negotiations process seems to get ignored.
The problem actually started with Jimmy Carter, who broke off communications with PATCO and began formulating a plan to break the union. Reagan promised PATCO that he would negotiate in good faith, resulting in PATCO being the only labor union to endorse him. Then he flipped 180 degrees on Jan. 20, and appointed two union-busting CEOs as heads of the DOT and the FAA, setting up the confrontation. He flat out lied through his teeth. He refused to negotiate in good faith, and began preparations to bust the union. That was the goal all along, and the only chance we had was to win.
And in the aftermath, he employed scabs from ALPA (the pilot's union), which was illegal, reassigned military controllers (illegal), and reassigned thousands of low level controllers and employees to positions for which they were not trained. All of this violated existing FAA regulations, and put millions of peoples' lives at risk. We were just plain lucky that there were no major disasters.
And, ironically, AFTER he fired the controllers, THEN he fast-tracked the stalled new radar system, which was the most important sticking point bringing about the strike in the first place.
This was the final straw in the demise of unions. You saw very few strikes after that, and organizing fell of dramatically. States began to pass right-to-work-for-less laws. 1981 was a watershed moment for unions.