I'll start by saying that correctly stating Kirchhoff's law is quite tricky. "Emissivity equals absorptivity" in a certain sense, but they may depend on wavelength, and angle of incidence (or emission), and polarization. In magneto-optic materials, you can have high absorptivity from one direction balancing high emissivity into a different direction!! (This is called "Violation of detailed balance".) Kirchhoff's law can be stated in a correct and universal way, but it's more complicated than you may realize.
The second point is that most serious mathematicians / scientists / engineers don't spend time reading 100-year-old papers, and don't rely on or expect these papers to be fully correct. Important results (results that people continue to rely on) get re-derived in countless textbooks, lectures, homeworks, etc. And as time goes on people usually find better ways to explain, think about, and contextualize these concepts than did our pioneering forebears. (It can be fun and useful to read old papers, don't get me wrong, but the pioneers don't get the last word.)
As a matter of fact, I have not read Planck's paper and as far as I know it may have errors in it. But I know several airtight proofs of Kirchhoff's law from modern textbooks and papers, and that's why I believe it.
I have, however, looked at the Robitaille-Crothers paper, and they just seem so totally confused about basic aspects of thermal radiation that I'm embarrassed for them.
Here is a typical example from their paper:
It remains an experimental fact that good reflectors, such as silver, are never utilized to construct blackbodies, in direct contradiction to Kirchhoff’s claim that cavity radiation is independent of the nature of the walls from which it is comprised.
I encourage anyone reading this to borrow a thermal infrared camera and take it into a room with reasonably uniform temperature and no windows / lights / etc. Let's say a bathroom, which has a nice piece of silver in it---the mirror! Look at the wall with the camera - it has a certain glow. Now look at the mirror - you'll see the same glow with the same intensity. (Isn't that obvious? You're just seeing the reflection of the opposite wall!) In fact, anywhere you look (except your own warm body) will have the same glow from thermal radiation. That's what Kirchhoff, I imagine, was saying. The bathroom is a cavity, and indeed everywhere you look, you'll see the same uniform glow with the thermal infrared camera, whether it comes directly from a wall, or indirectly from one wall after multiple bounces off other walls, or even from the air if it's foggy.
So you can (and should!) check for yourself that having some silver walls does not prevent a closed cavity from having a blackbody radiation spectrum inside. At the same time, using exclusively silver mirrors would be a silly way to make a laboratory blackbody source! (It's not strictly impossible to do so, it just entails working unnecessary hard.) There's no contradiction here, unless you're hopelessly confused, as the authors appear to be!
