The statute fixed that privacy issue and then put requirements on when wiretaps could be used by the government. Before this law and the Supreme Court decision the government was very... lax regarding warrants for wiretaps.
On the other hand, it's not a privacy violation for the DHL guy to voluntarily turn over material to the government and the government to use that as evidence against you. Think of the stereotypical "guy at the photo lab" who notices that someone dropped off child porn to be developed, you (sadly) used to see that in the news yearly, and no one thought anything of that being turned over to the government.
In some cases third-party services have privacy protections added by law. E.g. the cellular communications interception thing, it used to be very easy indeed to intercept a cell phone call just by sticking up an antenna. In 1994 the law was changed to make it illegal to intercept those calls in most scenarios, and require search warrants even for that "public transmission" of communications (but the law also required cell phone companies to make it possible to wiretap, the "CALEA" provisions).
Given how strictly regulated banks are I would be very surprised if there are not similar laws providing some semblance of legal protection to the contents of safe deposit boxes, but I'm not sure and don't have time to Google it.
Regarding phones, those also have specific laws attempting to protect wireless (cellular) communications from eavesdropping (with the exception of legal wiretaps, which cellular providers are required to be able to support).
Likewise for pagers, those have specific laws making it illegal to intercept the communications sent to pagers.
For normal phones, a specific law had to be passed to require warrants for interception of communication (the "Wiretap Statute" from 1968, later extended by the Electronic Communications Privacy Act of 1986).
So the point to all of this is that those privacy "rights" that you talk about are not 4th Amendment rights at all, they are protections granted by specific legal action on the part of Congress. Had that legal action not been taken then you'd be right back in this same "interesting 4th Amendment question" that we're talking about for this.
The bottom line is that if you're willing to give info to a 3rd party unencrypted you need to assume the government can be given access to the same information unless there are specific laws forbidding it.