[calf]

See, that's the thing. If you are saying Law of Excluded Middle matters for justification of using proof by contradiction, then we are suddenly really talking about the justification or not of classical versus non classical logics. That's kind of the author's point in the last paragraph of his article, that there's a metamathematical thing going on even if the student cannot quite articulate it. The real problem is not LEM's place in propositional logic but the cognitive move of hypothetical reasoning. Even the article leaves the question open ended.

To make this less abstract, note that in your own example you used a proof by contradiction to justify the technique of proof by contradiction. That is inherently problematic.