Suppose the interior of the cavity is made of a non-black body. Then in thermal equilibrium, the spectral radiance of the non-black body, say $R_T^*(\nu)$, will be different from the spectral radiance of a pure black body, $R_T(\nu)$. So the way I see it, inside the cavity there is a distribution of radiation with spectral radiance $R^*_T(\nu)$ but somehow when it passes through the hole it becomes $R_T(\nu)$ because the hole behaves like a black body? I'm finding these two things to be irreconcilable. Can someone expand on this please?
Also I have a more practical question. How is the cavity brought to thermal equilibrium in practice? From what I've read, it's the hole that really absorbs like a black body so to make it be in thermal equilibrium like a black body does one have to send radiation through the hole so that it absorbs all of them and emits at the same rate to be in thermal equilibrium? How would that work? Since you're already sending radiation in through the hole you can't analyze the emission spectra coming out of it too. Or is there actually a way to do it?