You're going to become fast friends with the postal inspector. They will most likely require you to maintain a list of every sender and receiver.
Ignoring that, let's compare this to the standard I don't want you to know where I live strategy: a post office box. Let's say a streamer or some famous YouTuber who doesn't want people to actually have their home address.
Anybody can send to a post office box, they just need the address. It's not entirely clear, but your service would require both the sender and receiver to have an account and relationship with you.
The privacy conscious person cannot take packages directly home from the post office, they need to be scanned for trackers, air tags etc, or opened in a neutral location. Your service would send directly to a destination address, so a single air tag would destroy all privacy, unless the destination was a PO box, but at which point what is the customer benefiting from?
I think it's an interesting service, but I don't see it working. The closest I would see it to is virtual post mail, or other virtual mailbox services. They scan mail when it comes in, and then email or reship that to another destination up to you. That's kind of the privacy arbitrage layer. Otherwise there's the post office boxes for people who want to receive without giving away their location.
If two people want to have a transaction without any third party knowing, shipping it via the post is always difficult. Labels are scanned at every office. And I think your service will quickly have tracking requirements put onto it, quite frankly your early adopters will almost certainly only be sending illegal material.