The presentation in the quote is awkward and doesn't work, as you pointed out, but Lee could instead have just mentioned that there's such a construction with q

The presentation in the quote is awkward and doesn't work, as you pointed out, but Lee could instead have just mentioned that there's such a construction with questions of the form "is the answer to the previous question YES [NO]?" which is not constant (not all 1s), and does have the property he wants, i.e. the are such sequences that cannot be specified finitely (corresponding to binary sequences with that property). I don't think you've shown that Shannon's theorem isn't a problem here so much as the argument presented did not successfully convey that it's a problem. Tangentially, I'm interested to see a formal statement/proof of the form of Shannon's theorem add used in this article, because I can sort of intuitively see why it makes sense but I'm not familiar enough with the technical details to guarantee that this is even an appropriate use of the theorem, assuming the yes/no question construction has all the properties Lee wants it to have.