Most attorney-client privilege laws explicitly exempt any communications that can't be expected to be private, which would presumably include the exterior envelope of a piece of mail, which is effectively a postcard (the archetypical not-private communication).
The other problem you'd have here is that you can't generally enforce the privilege, beyond excluding evidence in a trial.
This is an area where parallel construction[1] can be applied for evidence laundering. Usually these terms are used to describe unreasonable search/seizure in violation of the 4th amendment, but it can also be applied to evidence collected in violation of attorney/client privilege.
The evidence can't be submitted in a trial, but that's pretty much the only thing that can't be done with it--you can use it to obtain other evidence. In theory that evidence would also be excluded as "fruit of the poisonous tree", but in reality and violation of the constitution and basic human rights, law enforcement frequently uses parallel construction--showing an alternate path by which the evidence might plausibly have been obtained--to get evidence in front of jurors which they could not have obtained without violating the constitution they supposedly uphold.
I don't think any of this has anything to do with attorney-client privilege, which is the only thing I'm here to talk about. Postal mail metadata is about the least interesting possible thing to discuss; the privilege thing here was the only interesting angle I saw.
The Post Office's regulations for these mail covers do explicitly exclude mail "between the mail cover subject and the subject's known attorney." I agree that attorney-client privilege probably doesn't require this for "outside of the envelope" information, but the policy seems to be more of just reacting to the... vibes of the privilege.
