It may make sense if we think of this being tied to US companies. How would Anthropic/OpenAI force US companies to use their products only? They'd have to somehow influence the regulations to ban the Chinese models or any other open weight ones. It's about companies who spend tens and hundreds of millions on tokens staying and keep paying. Not you as an individual or a small startup. They want something like "These companies didn't pace the frontier so their use/output is banned in US" as the angle they want to play.
Anthropic saw that this kind of rhetoric can work. They stepped on their own rake, so to speak, when they drummed up the super-capabilities of the own model, and then didn't kiss the ring the right way and found themselves export controlled.