Sometimes they win demands even when the strike is illegal. The 2018 West Virginia teacher's strike turned into a wildcat (unauthorized) strike but that didn't stop them from winning a 5% pay raise. It does depend a lot on community support and sympathy.
A wildcat strike is always illegal in the US. Organizing a union isn't illegal in any state, it's about whether the union gets official recognition and official bargaining rights. My point was that it's possible to win even if you don't have those things.