Weapons inspectors are just as beholden to rich states as the WHO no? It's the nature of the beast unfortunately. Look at war crimes tribunals that only try Africans because they're the only ones weak enough not to resist them. I think what you say about the WHO applies to most UN organisations: they're only useful in poor countries that no one cares about. They try their best but they're hamstrung. Look at the Security Council. They can't get any kind of meaningful resolution through without US/UK on one side, or Russia/China on the other vetoing it.
Perfect shouldn't be the enemy of the good. I agree with most of what you've said there but I still think something is better than nothing, which is what is currently in place.