Some people seem to think the previous space race was about technology. That was all incidental. The Space Race was entirely geopolitical. It was a conflict proxy, an artifact of the Cold War. Any technology was entirely incidental.
The US has been talking about a return to the Moon for 50 years. George W Bush talked about it in 2004. It still hasn't happened. Artemis is limping along but it's entirely pork barrelling for the overly expensive SLS program that really no future.
Some might say SpaceX will come to the rescue. That's... doubtful. Notably, Elon calls the Moon "a distraction" [1]. Why would he do this? It's free money from the government.
The answer is actually pretty simple: Tsarship simply isn't designed for this mission type. Landing a Starship on the MOon is much more complex than, say, the LEM for the Apollo missions or the proposed Chinese lunar lander. If you could, your astronats would be 40 meters off the ground. The big advantage of a "traditional" lunar lander is it can't really topple over. Plus the Apollo LEM also had a very simple engine that could only ever be used once but the big advantage was that it was extremely difficult to fail.
If you exclude all that, Starship is behind schedule and still requires developing technology that they haven't even begun to test, most notably in-orbit refueling.
So why is China doing all this? I suspect it's mainly to develop their own reusable rocket program with a side of national pride. China is very concerned with their national security interests. Being able to launch things cheaply is a critical national security interest.
Still, the mission architecture of China's mission (from what I've read) is still fairly complex, requiring two vehicles to rendezvous in lunar orbit. That's also why I think the primary goal is orbital launch capacity.
[1]: https://x.com/elonmusk/status/1875023335891026324