I'm not a geopolitics expert, and I assume you're not either, so I'll just say what I feel. As an European, deep down my unconscious mental picture of the situation here is probably this: USA is a geopolitical and economic power, China is a far away country that assembles parts and devices for western companies. This mental picture is wrong and hilariously oversimplified (I know rationally that it's wrong), but this is the stereotype I've absorbed from my society.
If both counties actively tried to win, and China managed to build a Moon base before the US that would probably make a huge blow to that (subconscious) mental picture.
that's because the US and the rest of the west pretty much decided after 1990 that history had ended, even as their societies crumbled from the costs of the cold war.
the example of china's rise and eventual dominance disturbs that narrative but doesn't demolish it entirely. sinking a US carrier or building a moon base before white countries do would be concrete examples that they can't explain away.
> that would probably make a huge blow to that (subconscious) mental picture.
And?
Okay, so let's say this happens, and now some people think: "Wow China is more powerful in space than the US!".
What is the real world impact of this? Did Chinas army just get 10x more powerful? Does the CPP now own the moon? Did Chinas [economic challenges][1] suddenly disappear? Did their [demographic issues][2] suddenly improve?
No, of course not.
The only real world impact of this: China would now be faced with the choice between blowing billions upon billions of dollars anually for what is essentially a vanity project with little to no ROI, or find a way to abandon it quitely without too much public fanfare. And of course, the whole thing is constantly only one malfunctioning airlock away, from turning into a pile of dead astronauts and a complete PR desaster.
So please tell me, and you are completely right, I am not a geopolitics expert, in terms I can understand, what the specific and tangible benefits of building a Moon-/Mars-Base/NewMannedSpaceStation/etc. are supposed to be, in terms of geopolitics.
And sure, I can see some change to mental images, and yeah, that might e.g. attract some business that would otherwise be somewhere else, or make the odd contract negotiations go smoother. But at the end of the day, these advantages, such as they are, would still need to offset the pricetag of the whole show, and I don't believe they would.
If both counties actively tried to win, and China managed to build a Moon base before the US that would probably make a huge blow to that (subconscious) mental picture.