I think the key is that they also can't let Anthropic employees who are foreign nationals use it (e. g. overseas remote employees, people on H1-B visas or green cards, etc.)
That would probably make it very difficult to maintain and develop if there's even a small number of such employees, and I suspect Anthropic, who pays large sums of money for what they perceive as the best talent wherever they can find it, has quite a few.
And, if their best talent is anything like the other "leader in their field" people I know, they aren't particularly interested in becoming American citizens.
You're right and that is the issue, but I do want to point out that IIRC for ITAR purposes, US permanent residents are considered US nationals.
US vocabulary is confusing.