I get your frustration here, but keep in mind: your use of that word is not harming Stripe any more than alternatives you could use, but it does harm an unrelated and oppressed group.
It's not at all material to the point you're making, but to nitpick: Hans's first move was actually 1. d4 and thus the Wikipedia article you should be linking to is that of the Indian Defence [0], not Alekhine's Defence!
In their example, you're not giving S as an input to K — instead, you're using K to write another program P which iterates over all possible strings and returns the first whose complexity is some value ("2 million" in their example).
That program P will certainly be longer than K (as it contains K), but not much longer — it's adding to K only the instructions needed to iterate over strings and define the threshold complexity ("2 million"). P will then produce that "2 million"-complexity output, but it didn't need any input and thus the complexity of its output is truly just the length of P (which is smaller than "2 million"). It eventually stumbles upon S by going through all possible strings, and didn't need S to be provided.
The main idea of the proof is very similar to the interesting-number paradox [0] or the Berry paradox [1].