The convention does not prevent national law from providing private liability which may come into play between entities subject to the jurisdiction of the same state or between the state who is liable to other states under the convention and entities operating within the state. So, there is no need to update the convention; the states from which private launches operate simply need adequate domestic law to cover both fully-internal liability and private launcher liability for claims against the government under the convention. (And the US generally does, with the basic regulatory regime being adopted and the private space launch industry operating in the 1980s; it is not an issue that arose with Musk/SpaceX.)

FAA launch licenses require substantial liability insurance. 500 million in this case.

https://drs.faa.gov/browse/excelExternalWindow/DRSDOCID17389...

States can set whatever rules they like internally. The US can make SpaceX pay them back if they want.

No expert, but I would assume, the USA would front it, but then take a case against SpaceX. So it would be Boat Owner v. USA, then USA v. SpaceX shortly after. Although I could be totally wrong.

But yea, seems appropriate to update it or if that is going to be the process, write it in stone.

Every rocket flight has to be approved by the government. No launch until FAA (and also FCC) OK's it.