Impulse responses can only represent linear time-invariant systems. Like delays, reverbs, equalization curves.
Distortion is non-linear, it is something like a max(-1, min(1, input)) function (a waveshaper, like you said), and it produces harmonics when applied to audio signals.
However guitar pedals also have some additional circuitry to "sweeten" the distortion, removing the extra harmonics added by the clipping diodes. Tubescreamers are notable for cutting bass and enhancing mids. An IR is able to capture this. This is important for guitar pedals, and the reason multiple of them exist.
If you capture the impulse response of an overdrive pedal you'll be capturing only the frequency response of a distorted impulse. If you process clean guitar trough this you'll simulate the frequency response but not the distortion itself, so it will just be a clean guitar with a tinny, shrill, sound, not an overdriven guitar sound.
One way around it (other than the idea in this article!) is doing multiple passes of Impulse Response capture with different amplitudes, this will capture this distortion non-linearity. This is supposedly how a Kemper Profiler works.