The ICC is overwhelmingly popular with most countries and people, most people who oppose it are those from countries which harbor war criminals and try to make their opposition look like it stems from legal issues with international jurisdiction, and not the truth which is to protect their country's ability to commit war crimes unabated.
We don’t have good enough contemporary global polling to confidently say that a majority personally supports the ICC.
However, I suspect you’re right that most people probably support the idea of the ICC but countries choose not to be part for a variety of reasons, other that “harboring war criminals”.
The following countries are not party to the Rome Statute, which is what established the ICC:
United States, China, Russia, India, Israel, Indonesia, Pakistan, Turkey, Saudi Arabia, etc.
ICJ and ECJ seem fine. I agree that the ICC is toothless (unless it’s investigating Africa for some reason). But if countries agree to its jurisdiction that’s basically glorified arbitration. The Philippine with Duterte show the benefits to a population to submitting to jurisdiction. I bet a lot of Americans being abused by ICE wish we had a check on our political leaders outside our own legal system.
Sure. You know what, I don't know enough about it to argue if it is. What I can say is it isn't toothless. International treaties can create robust judicial systems.
Saying the security council works is... A perspective.
During the general assembly there were plenty of comments as to how it's not representative at all. There are 5 members with veto power (us, Russia, uk, France, china), why these? Because they won the WW2? The whole southern hemisphere doesn't hold veto power.
The security council works very well, if you're one of the 5 permanent members.
The first "supranational" court system was created with the USA fully on-board: https://en.wikipedia.org/wiki/Nuremburg_trials
In fact, they liked it so much that they re-enacted it at home: https://en.wikipedia.org/wiki/Subsequent_Nuremberg_trials