This is really trivial in the end. Inconsistency is "Φ ∧ ¬Φ", law of excluded middle is "Φ ∨ ¬Φ", so in the assumption of law of excluded middle is hidden the assumption of consistency. That's what he means by
The above theorem means that the assumption of consistency is deeply embedded in the structure of classical mathematics
Edit: I agree with you in the end, the proof is valid, but the conclusion does not hold in A, but one "level" above, so it does not contradict Goedels theorem. Am I reading this right?