Why don't you think the political reasons are enough? Political reasons were why the US did it originally right?

Maybe political is not the correct word. The US did it to prevent existential threat from Russians (this is my understanding) and I don't see one here.

What existential threat was addressed by sending humans into space?

Putting humans into space is a roundabout way of letting people know you can put anything you want into space. That is, it's a demonstration of superior technology, which usually means superior military capabilities. In the context of the Cold War, it is reasonable to assume that both the USA and USSR feared that the other might become overconfident, underestimate their potential enemy's defence and make a first strike.

Therefore, the logic goes, each side needed to frequently show off their advanced technology whilst avoiding showing any secrets: ostensibly civilian space exploration serves that purpose rather well.

Putting people into space didn't stop the Russians from doing anything that was an existential threat to us, though, did it?

Possibly it did: the Russians might have thought that they were sufficiently ahead of the USA to start a war and survive it. Perhaps, in a flare-up of nationalistic sentiment combined with a bit of political instability at home, the risk of mutually assured destruction wouldn't have seemed too high. In this hypothetical scenario, seeing pictures of American spacecraft landing on the moon, astronauts doing spacewalks before making safe re-entry at supersonic speeds might have made the notion of surviving a war seem untenable, and would have put the Russians off the idea of a first-strike.

I've read a similar argument for spying - that countries begrudgingly want a certain amount of espionage to take place in peacetime. This is because it's better for everyone to know the extent of each other's military capabilities than to accidentally start an arms race out of a misplaced belief that their rivals are suddenly increasing development of weapons.

I was born in the post-USSR world, and am also British rather than American, so perhaps take my perception of the Cold War with a pinch of salt. :)

The idea that seeing the effects of a nuclear weapon wouldn't be enough to deter a war but seeing a man walking on the moon would is absurd. Is there any evidence for it?

I don't have any evidence, unfortunately. The Wikipedia page for the Space Race mentions that it was considered critical for the national security of the USA, but again there's no citation as far as I can tell for this claim.

So... politics?

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.

Why are you asking about a scientific reason?

I think the question may be:

What significant difference does it make if we put a woman (or woman of color) on the moon first vs putting another white man on the moon. [with modern technology].

I could think of a few positive reasons to do this, but it shouldn't be the main driving force of competition.

No scientific difference. It's political. A president can trumpet it as a great achievement for humanity. Helps with getting funding & public support, yada yada.

It feels like the value in political virtue signaling is quite past its peak, in fact I think there's something of a negative value to it in a lot of important circles.

The only scientific reason for sending humans to space is to develop better technology for life support on longer missions. At this point automated probes can accomplish most other scientific purposes better. So yes, you send people with different physiological characteristics to further that mission. If you’re sending people for non-scientific reasons than you do it to be first.