I believe the Gaon has a piece discussing this. (And perhaps the Dubna Magid as well, not sure).
And I think one can ask more than that. You were asking “al pi chesbon, he was right”. But even if every step of the line was wrong, and the chesbon didn’t work out, and the Malach Hamoves was found to be WRONG — What is the concept of a Malach being wrong? A Malach has no strengths of his own; he is a conduit of HKB”Hs power. The Malach is simply a messenger to carry out HKB”Hs Will. The Malach has no agenda of his own. The Malach HaMoves was created by HKB”H to fulfil a tachlis, and is no less than Malach Gavriel or Michoel. The same is with the Yetzer Hora – they/he are/is just doing what HKBH commanded them. So how can that be wrong??