There's the obvious meme of "the US used to be able to do it, but can they still do it?". That wouldn't stand in question if the US had say a Mars mission, but if all the US can show are some low earth orbit activities while China has astronauts walking the moon that makes for a great propaganda point for the Chinese. Something to the tune of "As the American empire declines, the Chinese empire rises"
But the more impactful point is that the Chinese don't want to stop at what the Apollo program accomplished. They want to build a moon base, turn it into a lunar research station and invite other countries to cooperate. If the Chinese are wildly successful on that front, cooperating with them to get access to their moon base might be very enticing. Both for research about the moon and about low gravity. If the US doesn't answer with their own moon base that might end up in a reversal of the ISS situation (where everyone except China was invited to cooperate on the ISS).
Of course we don't know whether the Chinese will be successful in those points. But so far their space program has a great track record. They did manage to build their own space stations and lunar rovers, everything after that is, as you say, mostly a funding challenge