Some very good space lawyers disagree with you. Not all, but this is an area of strong debate. (The Moon Treaty was never ratified by any nation with an orbital space capability - I presume you mean the Outer Space Treaty). And what if a corporation is incorporated in one of the several dozen nations that have never signed, or have signed but never ratified?
The FAA presently administers space flight to/from the US already - if you want to launch a rocket with more than small-hobby capability you have to get a permit. You may also have to get NASA to sign off on the equipment to get that permit. So FAA is offering to extend this to provide a reasonable alternative to the 'Wild West' for at least those companies with some US component - residence, launch, tracking, lots of other aspects, IOW almost every rocket flight from almost any country. IMHO it's a reasonable offer.
I foresee other nations establishing a cooperative agreement for each of their aerospace agencies to cooperate in this, taking responsibility for their own parties and working together. In general cooperation has been the case for most space operations - viz. the continued launches of rockets from Russia, carrying satellites from the US and elsewhere despite the various international goings-on. This type of international cooperation is analogous in some ways to the way that patents and other IP rules have gradually evolved to a modicum of international normalization.