It seems that the spline visualization only works for degree 2?

It should also work with p != 2 (tested with chrome and safari). Which browser are you using?

Chrome on macOS (retina/hidpi display). p=1 produces NaN paths, and for p>2 the curve isn't drawn for all control points. For p=3 this is what I get: https://imgur.com/a/8fhpUSy

(It could very well be that I misunderstand how higher-degree NURBS are supposed to work, though)

Seems to be correct. Check and modify the knots in the parameter space. E.g. you need a knot multiplicity = p to interpolate a control point.