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.