It doesn't have to be up to them, and by no rights should it be. If the US government understood what was happening, OpenAI et al would be shut down overnight.
I'm just some guy, and one of many doing grassroots lobbying about AI extinction risk. And yet, if there had been just 200 more people having my very modest level of impact on US lawmakers over the past 2 years, I think we would have a global AI treaty today, or be very close.
Other countries have less impact, but far from zero! The impact of multiple middle powers formally recognizing that superintelligent AI is an extinction level threat, may be very close, and should be banned would be very substantial.