It's a principle that the person who presents a positive case is the one who needs to come up with the evidence. I can say "there's no god" without evidence, it's the guy who says the God of the Bible is real who has to explain what he means.

> I find your claim...

Note the reply chain, I didn't claim anything.

> By this logic, if I say something like “there was no moon landing”, I could demand that you offer evidence to the contrary instead of offering any of my own.

The contrary would be that there was a moon landing, and indeed the person who claims it happened would be required to come up with the evidence. Luckily there's a lot of it.