The grievance you express here basically comes down to "Who decides?" The reason the ICC is objectionable to the U.S. (and China, and Russia, and India - there are a bunch of other non-signatories of the ICC agreement, and I'm not picking on the U.S. here, more trying to illustrate a dynamic) is because it sets up a justice system that is not accountable to the U.S. political process.
The reason I'm saying that this undermines U.S. soft power and legitimacy is because it shouldn't matter who decides. If your country actually stands for freedom, justice, and universal human rights, then another human working from those principles would reach the same conclusion you do. That's what makes them universal human rights. So the implicit message when judges are sanctioned because other nations disagree with their right to make decisions is that maybe those rights aren't universal after all, and that some people's rights are more right than others. It's telegraphing insecurity - someone who knows they are in the right will happily let others make the decision, because they trust that others will come to the same conclusion as them.
The rest of your post also comes from that frame. You may actually be right - I acknowledged that neutrality and objectivity is likely a fiction. The thing is, that fiction underpins the ideas of rule of law, democracy, freedom, justice, and peace. All of those stem from a basic trust that your fellow humans, once they are trained to think in those terms, will come to the same conclusions that you do. So yes, perhaps all of international relations is power players advancing their own interests - but once everybody believes that to be true, we will lose rule of law, democracy, freedom, justice, and peace.