One of my pet hates in life was seting up string lines , simply because no mater how much tension I put on it . Gravity would always make it sag slightly in the middle . Now not once did it ever convex. So tell me, if a string line wont convex & it actually concave's due to its weight & fall. Why on earth would you think the surface of the earth would act the opposite to everything else. ?
I finally found your string line "experiment"!
Your string line sags simply because it is only supported at the ends and gravity pulls the line down. An extremely high tension is required get a very small sag - yes I know the problem, especially when a line level is hung in the middle.
But, would you expect bodies of water, like swimming pools and lakes, to be concave? Small bodies of water look flat, but
to me very large bodies like the ocean do indeed "convex".
These and earth's surface are not concave because they are supported (more or less) uniformly all across their surface.
There are examples of rock bridges, but these are rigid and are always too small to show curvature (or not) from the earth's shape.