Or in the case of my recent Euler explorer, minimizes the L2 norm of tangent error. That said, most of the error norms will give similar results, as long as you're actually minimizing something. You're absolutely right about this though, the main reason people pick that one is that the math comes out easy, it's just a bit of trig.
Or you could describe the quarter circle exactly using a rational spline.
A more traditional choice might be the cubic curve where the maximum distance between a curve point and the circle is minimized. Or it might instead be desirable to preserve the arclength of the circle and minimize curvature deviation. Or it might be desirable to preserve the circle’s area. Or ...
Depending on the application different choices might be better. For some purposes it is not strictly essential that the endpoints of the curve directly fall on the circle.