I don't see how you get that reading. Anthropic is clearly allowed to sell Claude to companies not doing business with the US Military. If anything that's more likely to be non-US companies.
Because Anthropic sells Claude through other companies that in turn do business both with Anthropic and the government. These intermediaries, large cloud companies, can't offer Claude anymore if they want to keep the government as a customer.
Sorry, the "they" referred to the hyperscalers
IIRC, the supply chain risk designation is sticky which is why it tends to ultimately mean "nobody can work with this". Amazon using claude means a DoD company can't use Amazon. Every business that touches claude gets tainted.
It's a bit like how the US Cuba sanctions worked and why they effectively isolated Cuba from everything.