> It also is in Schwarzschild coords

True--indeed, the statement that it's a spacelike line is an invariant, independent of any choice of coordinates. But it's a lot harder to see that in Schwarzschild coordinates.

> Kruskal-Szeres really is only useful for demonstrating that the event horizon is not such a weird place, but it does nothing for the singularity at the center.

I disagree, and I think many GR physicists would as well. Indeed, both classic GR textbooks, Misner, Thorne & Wheeler (1973) and Wald (1984) discuss Kruskal coordinates and their associated spacetime diagram (MTW in Box 31.2 and sections 31.5 and 31.5, Wald in section 6.4), and those discussions contain a good deal more than just showing that the horizon is "not such a weird place".