When I say that strikes work, I don't mean that unions get each and every last thing they ask for. That's an unrealistic goal in any negotiation. I mean that strikes allow unions to get things that they would not get without a strike. This is primarily because a strike adds a very powerful stakeholder to the outcome of the negotiations: the public. When negotiations involve only workers and management, management is often able to simply say "fuck off." Management can wait them out—workers will run out of money and start starving long before their managers do. If managements feel that they can save money in the long term by telling workers to fuck off with their contract demands, they will do it, even if it means taking a financial hit in the short term. This is the cold logic of capitalism. Absent any direct incentive, management will always take a dollar out of workers' pockets and put it into their own, if they can.
A strike, though, acts as a check on that imbalance of power by inviting a very powerful third party to the table. Rahm Emanuel, perhaps, would be happy to tell teachers to fuck off. When Rahm Emanuel has a million angry parents calling his office demanding that he fix the god damn teacher's strike so their kids have somewhere to go all day, things change. Mike Bloomberg, perhaps, would be happy to tell the NYC subway employees to fuck off. But when they go on strike and the subways stop running and the entire commuter-driven metropolis grinds to a halt, eight million people collectively demand a solution, and fast. Verizon would surely be happy to tell its employees to fuck off and take what it gives them. When nobody can get their cable fixed in time to watch the game, Verizon will feel the wrath of the world, pressuring it to find a solution. The public is awesomely powerful, and self-interested. The public wants things to work. The details of how that's accomplished usually get drowned out in the primal scream of "fix it now!" This pressure mostly falls on management. Sure, people get angry at the unions, but unions, excluding corrupt ones, are not primarily concerned with PR (at least not to the extent that corporations or politicians are, by necessity). They're concerned with improving the lives of their members. They are the only thing standing between workers and the "good will" of management, which is often the same as oblivion.
Are strikes an inconvenience for the public? Yes. That is why they work. And that inconvenience, in the long run, is a small price to pay for living in a country that respects freedom enough to allow its workers to organize. Strikes are the pinnacle of workers exercising their freedoms in this capitalist system of ours; conservatives should love them. On principle, it scarcely matters whether the workers are public or private. As John Cook wrote about Chicago's teachers, they are "participants in a labor market. They are free to organize and to withhold their labor if they don't like the deal they're getting. They will either get what they want, or they won't. This is how things work."
Restricting the right to strike is tantamount to forcing people to work against their will. That's an even more onerous government demand than taxes. You would think the Republican party would be protecting workers' rights to strike at all costs.
Some countries with far more radical economic histories than ours can find themselves paralyzed by frequent strikes, to the detriment of the nation. We're not them. We're not Greece, and we're not Venezuela. We're not even close. We're America, where "socialism" is still considered a pornographic word in politics. The working people of America—which is to say the majority—would be better off with more strikes, not fewer. Because they work.
Of course, in order to have strikes, we need unions. That's another thing to work on.