The most obvious is that any international body would be easily controlled by the big players, so you'd end up with more centralized control by the same national entities, but now they'd be controlling other countries launches as well.
The other problem is that lately international organizations have a pretty bad track record. Two examples, which I've chosen because they are actually both very important incidents and also squarely in the domain of the respective orgs: WHO with Covid with a mostly useless and visibly politicized reaction; and UN with Gaza, with a large block of Arab voters who are basically stuck at condemning Israel, but systematically refuse to actually step up and help with the problem. Both incidents are literally what those orgs were created to handle, and yet they don't.
Also space launches have a military component, not always public. I doubt many would agree to let an international body poke their nose in that.