Having said all that, sending cash through the mail just feels like a really obvious money laundering vector. You might claim that there's no distinction between that and, say, handing over a bunch of cash directly to a person, but in the world of money laundering each extra layer makes the operation safer!
I don't think that from this it follows that we should be opening envelopes all over, but there's a universe where the seller of jewelry is very aware that they are taken illicit cash, and in a a more just universe there's some sort of warrant.
I do not know how to square the privacy question with this, but I do like the idea of money laundering not being trivially doable.