You are right, the way you want to reproject is wrong.
Set Layer CRS does NOT reproject your data, but assignes the new CRS to unchanged coordinates, making the world 360x180 meters large.
Save As... is the right way, but only on the original data, not on the corrupted data from the previous step.
The world map from natural Earth includes the south pole. That point can not be reprojected to World Mercator, because the poles end up in infinity with that projection. So you have to cut off the bottom line, or remove Antarctica if you don't need that part of the world. Since ne_10m_land is a multipart geometry, you might have to convert it into singleparts before you can delete the Antarctic.