The general US population considered Sputnik to be an existential threat. The original space race had all the impetus of a (cold) war effort. China putting a w

The general US population considered Sputnik to be an existential threat. The original space race had all the impetus of a (cold) war effort. China putting a woman on the moon would be a little embarrassing to the US, but people would forget about it in a week or two. It wouldn't prove China's technological superiority, just some vague sense of moral superiority. And there are a lot cheaper ways to send that message.