"Why would other countries, that don't share the same anxiety about China as the US, would be troubled with the this?"

It's the other way around.

There is a high likelihood that many countries of the "west" (the "global north"?) will outlaw, restrict, or otherwise control LLMs and the tools that enable them.

The US, however, is blessed with the first amendment which makes it extremely difficult to restrain speech in any form - including code.

The US might pressure them?

Well, many EU countries like Italy and Germany officially freaked out about DeepSeek, ordering it to be banned from app stores etc.