The comments in this thread are interesting, and I think that all the reasons brought up as to why it won't work are accurate.
That said, I think there's a slight chance of value in making this a really premium service. Don't try to make it less expensive than USPS, make it lots more expensive than FedEx.
Find the people who like to drive. Pitch it as having an adventure. There's an old movie, Bed of Roses, where Christian Slater is a former lawyer turned flower delivery guy, because he likes seeing people's faces when they open the door and see the bouquet. You're not delivering a package, you're meeting a stranger and taking a package on a secret mission! Don't make it anonymous, have people meet in coffee shops right before they drive off to feel each other out. It's a daytrip outside of people's routine lives.
The people buying the service are people who hire traditional couriers, like bike couriers in cities, but for longer distances, when it has to be hand-delivered and you can call the guy on his cell phone and such. It's a premium service.
And then there's even a tier above that. A really expensive alibi service, where you pay someone to drive somewhere far away and have something mailed from a traditional Post Office there, to get the far-away postmark.
Good luck!