mathematician and educator here. A full answer to your question gets a little bit into the actual content of the essay, beyond just English language and usage issues. So, I'm not sure this is a great Q & A for this specific site, but I'm already here, so ... I'll answer your question.
English language & usage aspects: The sentence you quoted in your question essentially says: "I am writing against issue X. I believe that example Y supports my view. However, many people will use example Y against me, to support issue X." That is what is so "vexing" to the author: the common use (to him, misuse) of a specific example (namely, High School Geometry) to try to rebut his earlier points in the essay.
Mathematical aspects: (Caveat: I happen to support most of the author's points and regularly share and recommend this essay to others, including my own students.) Immediately preceding the quoted sentence in that essay is a section about what the high school curriculum is and should be about. It concludes with a hypothetical dialogue to illustrate the author's point that mathematics should be about proofs and demonstrating why facts are true, as opposed to just stating what the facts are and moving on. For many students, High School Geometry is their first introduction to actually writing mathematical proofs, constructing arguments that demonstrate why a fact is true. So, some people would naturally say, "Isn't that good, then? Shouldn't we keep teaching geometry as we do in high school, because you want students to learn about proofs?"
However, the author goes on to describe how, in fact, the way high school geometry is typically taught, it totally ruins students' understanding of how to create and share mathematical ideas. The methods are so formulaic and uncreative that they end up obscuring the practice of mathematics and turn students away from it forever. (I'm really paraphrasing, but that's the gist.)
I hope this extra detail about the content of the essay, regarding mathemtics and the teaching thereof, helps you make better sense of the English language issues.