Well a proper union to begin with would actually be the result of freedom of association. In China they are not as the Chineese workers don't have freedom of association. The union in China is just another part of the governments control apparatus.
And yes, I know there are and have been problems with unions in the USA. But also we don't have nearly the same problems in non-Anglo-Saxon countries (most notably Sweden where I live). In Sweden even the armed forces officers are unionised. It doesn't get more public sector than that and that hasn't led to any major problems. (Or differences in outlook compared to their NATO brethren. )
No, the difference is between, on the one hand, partisanship above all else, and on the other, an understanding that we are, at the end of the day, all in the same boat. Given the relative sizes of the US and Sweden, and the relative difference in absolute strength (military, economic, etc.) that we are more attuned to the latter way of thinking. So, that we have "powerful" unions (our current prime minister is a former union leader, that's his political background), doesn't mean that those unions can "destroy" anything, as that destruction will leave everybody worse off. They're as constrained by facts on the ground as everybody else. That's not to say that there can't be fundamental differences in how to order society though, there are, and that we have a large portion of "socialist" (not really) policies is of course a large part of our being in the top of the quality of living indices for multiple decades in a row. Our much stronger and better organised version of the UAW notwithstanding. (Again our prime minister began on the national arena as the head of the Metal workers union, the largest and most powerful of the trade unions, which he later became the head of.)
Now, I don't understand the "massive conflict of interest" that would supposedly occur when government workers are unionised. As I read your FDR reference, he was basically against the right to strike for sensitive positions. That's a given, whether they are government workers or not, you can't have your ambulance personnel go on strike. The same with the army of course, and indeed while the Swedish officers are unionised, they don't have the right to strike. Turns out you can still be effectively organised and have real bargaining power without striking at the drop of a hat (even without the right to strike). A strike is the nuclear option, and there are still plenty of ways to wage war without it.
So if your point is that unions won't work in the US. Sure, I don't believe it, but it's at least an interesting topic for discussion. But if your argument is that "since unions don't work in the US, they can't work anywhere", then hard evidence flies directly in the face of that, and I won't be moved one mm by it.