ICC doesn't have jurisdiction for what states do within their own borders - every court has jurisdiction. If you don't know them or have your own feelings about them, then that's your own misconception of what the court's function is.

In the end, even the creation of the ICC in 2002 was contentious because of what states viewed as infringement of sovereignty. It's a very new international body that still doesn't have the support it needs. Additionally, it's judges and prosecutors are financially sanctioned by the US (for investigating war crimes in Iraq and Afghanistan) which impedes their work.

It's less about being toothless, and more like states purposefully attack it and try to weaken it - meaning it does serve a purpose (which is why it's attacked) and can serve an even greater purpose if unimpeded.

So instead of blaming the ICC for doing their job in the quagmire that is international politics and the limited tools they have, you should be directing your ire at the states who don't allow it to function as it should and ask for more protections of the court.

To answer some things

• Bibi (et al.) and Putin are wanted by the ICC and will be arrested should they travel to signatory states.
• The ICC isn't part of the UN. You're thinking of the ICJ.

I don't know what you mean about legitimizing. The court functions as any court in the world and the prosecution needs to be able to prove beyond a reasonable doubt. If the court makes judgements outside of what can be proven or available evidence, then it's not a court and would not be legitimate (and probably be attacked even more than it is now).