Why does the chromaticity diagram look like that?

The xyY colour space is designed such that the colours of light you get by blending two points all lie on the line between the two corresponding points. This makes it extremely helpful when you want to figure out which colours you can make with a particular set of primaries. Similarly you can draw the colours corresponding to pure wavelenghts and figure out the entire space of physically possible colours by taking its convex closure.

These features are not really replicable in any other colour space, at best you can use a linear transformation of it (which XYZ already is, and it has almost all properties you could want of a choice of basis).

That's true. And I will add that the viewable color gamut of a display can be depicted with a simple triangle on the xyY plot. All you need to know are the three chromaticity values for the reg, green and blue phosphors — they make up the three corners of the triangle.