I understand your argument, in the highlighted part, but from the US perspective, such a situation would be exactly what they would want, since the Military would be under tremendous international pressure after such a coup, and therefore my argument that the US is perhaps trying to force the PA's hand here in making such a mistake ...
Yes, from the view point you are offering, it would be a quick, and total win for the US, provided - the sole purpose of the US is to turn the PA into a weak, defunct army overnight (They may desire so in long run, but not too quickly).
However, the disclosure of the memo would make the PA apprehensive about taking on the government. If the US really wants the PA to get embroiled in domestic politics, all along when it is warring with insurgents on the western border already, then why would they alarm the PA by acknowledging the existence of the memo? It would work against their (US's) objective of letting the PA go ahead with the coup. I mean, the PA is not made of fools who would just get irritated and angry at the government w.r.t the memo, and
even after the exposure, they would exercise the coup d'état.
Another point to note is that, (it is basically my understanding) - If the PA does somehow successfully take over the government, then it will not only become extremely unpopular (popularity ratings must be real low because of the series of incidents that took place from May 1 to Sept 20), it will also be overstretched. The question is, would the Americans, or anyone (esp Indians), ever want an unpopular, overstretched, and weakened Army overlooking the nukes? The hardliners (extremists) within the Army might want to ride the power their own way.
I personally don't think the US, Pakistan, or any other country, would want such a scenario.
Edit...
As Cyril Almeida and some others have pointed out, the military was completely on the back foot in those days, and facing a lot of criticizm, and many commentators were surprised that the GoP was not acting to take advantage of the situation and undercut the military's influence - now we know why - Zardari and his coterie were to busy with paranoid conspiracy theories and treasonous requests for foreign intervention in Pakistan.
Had to add this part of yours. Cyril Almeida is very precise and very correct with the analysis that the military was completely on the backfoot. If there were any ideas of coup going on, why would A. S. Pasha voluntarily present himself to the Parliament, and even offer his resignation? (Unless of course, there's some strong differences going on between Pasha and Kayani, as mentioned by Retd. Gen. Talat Mansoor.)