I think the round/flat Earth question matters, because if it's flat that opens up a whole range of existential questions like:

• Is there an edge, dome or what's beyond this place we call Earth?
• If we've been put here inside a dome for example, then:
  • Who did it?
  • Why?
  • How to get out?
• Etc.

In regards to "the edge", as you mention weirdheadaches. Take a look at the supposed flat Earth map if you haven't. You'll notice that Antarctica isn't represented as you would expect, but it's ice encircling all of Earth.

The funny thing is, there's a ton of strange things about Antarctica, e.g.:

• A treaty called (The Antarctic Treaty) which was signed back in 1959 (53 countries has signed the treaty by now). I believe it's the only treaty which haven't been broken for that long a time. Take a look into it - for instance it says:

  "No acts or activities taking place while the present Treaty is in force shall constitute a basis for asserting , supporting or denying a claim to territorial sovereignty in Antarctica or create any rights of sovereignty in Antarctica. No new claim, or enlargement of an existing claim to territorial sovereignty in Antarctica shall be asserted while the present Treaty is in force."

• You can't just jump into a boat and go there to explore it for yourself. People have actually tried this and it's better guarded than you would think and there's only few places you would be able to come on land, since 38% is made up of an ice wall, 44% floating ice, 13% ice streams and outlet glaciers and only 5% rock. You might however be allowed on a guided tour with strict limitations.
• No planes uses it as a shortcut.
• Before the treaty mentioned earlier, there was a military operation called Operation Highjump led by Richard E. Byrd. You might find it interesting what he had to say about what's "on the other side of the South Pole", as he talks about land the size of USA never touched by humans:

Again, I don't say the Earth is flat, but it's stuff like this that makes me find this theory very interesting.

Just a little thing I also find funny, if you take a closer look at the UN Flag: