Your
"The article is not about 'give us the code instead of a paper', but instead about 'give us the code as a supplement to the paper'."
is an appropriate interpretation of the paper. But I can't have much sympathy for that interpretation because the code can be so darned long and obscure.
Some 'pseudo-code' of, say, a dozen lines as a 'figure' is okay. In nearly all cases, any code very close to something that actually runs in practice would tend to be too darned long.
The 'knowledge' of the paper is supposed to be clear just from the paper. Then the code is supposed to be 'routine'.
If the main body of the paper does not make the knowledge clear, then including a few hundred lines of code has little promise of improving the exposition.
There's a good history supporting my point in books by Knuth, Ullman, Sedgewick, the paper
Ronald Fagin, Jurg Nievergelt, Nicholas Pippenger, H. Raymond Strong, 'Extendible hashing—a fast access method for dynamic files', "ACM Transactions on Database Systems", ISSN 0362-5915, Volume 4, Issue 3, September 1979, Pages: 315 - 344.
and more.
For your
"The basic argument is that sometimes text alone is not sufficient to explain - a point you concede to by mentioning that the formulae exist in textbooks along with the description."
I believe you are asking too much of the equations, not a lot too much but too much.
In simple terms, even pure math is, usually is, at best is, is supposed to be written in complete sentences in a natural language, say, English. E.g., in a physics book describing Newton's second law, we discuss force, mass, and acceleration, explain that force is a vector F, mass is a scalar m, and acceleration is a vector a, say that force is the product of mass and acceleration, e.g.,
F = ma
So, by the time get to the equation, almost don't need it. Of course, not ALL writers write this way!
A long sequence of algebraic derivations is supposed to have a start much like what I gave for F = ma and after that is just algebraic derivations, often with not much need for more text.
For
"Take for instance many protocol standards: they come with a reference implementation, because sometimes the text is harder to understand when you can't compare it to code which actually does what is being described."
yes, there is a practical challenge there. Finally the Unix world decided to 'go empirical' and have 'code runoffs' or whatever they were called where people just got together to see if the actual code, say, the TCP/IP stack with SSL, would actually appear to work. Of course, that empirical approach was always risky because it is a long way from a mathematical proof of correctness or even a solid engineering document. And we've seen problems, e.g., with security in DNS.
For more, the AI world decided to f'get about 'theory' or even usual scientific documentation and make the criterion just 'does the code appear to work'. This, too, has proven to be risky and, of course, is a big step back in documentation quality from good applied math, mathematical physics, and engineering of past decades. E.g., how the heck to know that the Manhattan Project was on track before the first test? Sure: There were a LOT of derivations and calculations; that is, the project didn't rely on just the empirical approach. Indeed, the uranium bomb was dropped on Japan as the first 'test'; the engineering was so solid there was no doubt it would explode.
Sure, there is a big role for testing, but, still, the 'knowledge' in a journal is supposed to be able to stand on its own without much help from detailed documentation on code, electronic circuits, mechanical mechanisms, or chemical reactions. In particular, in experimental science, the published paper is supposed to be clear enough to permit others to reproduce the experiment, and to do so without detailed photographs and drawings of apparatus.
When I was an undergrad, one summer I worked in the lab of a famous scientist. He explained to me how a scientist is supposed to keep a good notebook and really good records. Then if another scientist wants to check details or reproduce results, sending the notebooks and data is supposed to be an obligation, and the materials are expected to be handled with great care and then returned.
So, yes, at times, in practice, there can be more that is relevant than just the published paper.
Still, communicating the 'knowledge' of a research paper with a few KLOC of code is a bit much!