Interesting observation. That's simply part of the usual definition of a magic square, and it does indeed feel arbitrary once you compare it with the hexagon case, which is very symmetrical. I first learned about magic squares from math books as a kid, about 25 years ago, and just accepted those rules as given.
Upon a quick research, there are many variations, and the closest to what you're describing is the "pandiagonal" magic square, where additional diagonals are considered (except they wrap around the edges of the square in a slightly funny way, so that every diagonal still contains exactly N numbers): https://en.wikipedia.org/wiki/Pandiagonal_magic_square
Why not every line of knights moves too? Because the cells aren't adjacent I'd say. (Both immediately make the problem unsolvable since all corners must match).
Seems "unfair" that hexagons have multiple line lengths to consider. I think this article's modification is a good one in that framing: Shifting every number up or down doesn't make the magic squares any easier, but it certainly helps with hexagons.
(In the hexagons, all lines are considered even if they don't have the maximum length)
(PS: make sure you hover your mouse over the diagrams)