If the Earth was flat, then everyone would be experiencing night at around the same time.
Since the Sun would be visible to everyone on earth on the horizon the moment it appears on one site of the "disk", then when sunrise happens in Tokyo, the some guy in Paris would see the sun popping out from the east too. But that is clearly not the case because the sun is perfectly visible in the middle of the sky on one side of the world while the sky is completely pitch black on the opposite side.
Furthermore, ancient civilizations having a "less" informed worldview doesn't actually prove anything against the quality of their scientific findings. The Ancient greeks proved the curvature of the earth mathematically, using the same math that we use today, and if you do that same math you will get the same result no matter what era.
The reason you don't see curvature in the horizon is because the earth is so big that it looks flat from up close. For example, if you increasingly give a polygon more and more sides, it will slowly become more round until you can barely even tell that it has individual sides--but if you look at each of those sides from up close and ignore the rest, it'll look flat. If you ever see a ship sailing off into the horizon, notice how (even before it becomes too small to see) the bottom part of it starts to get covered up by water until it looks like the ship sunk into the water. No it didn't sink, it just got obscured by the curvature of the earth