In NYC at least (admittedly an outlier in the united states), most apartments don't have doormen and there's no place to leave packages and package pickup locations are almost uni formally in warehouse districts/boroughs hard to access via public transit
I actually had thought of a variant on the idea, but never implemented it. I love it, and would like to have it implemented here (maybe if things are going better for me next year, I could partner with them :) ).
USPS won't take UPS or FedEx. UPS and FedEx don't have a way into my building. I don't have a car to get out to the airport where the UPS center is. University mail room policy is no personal packages at work. So I basically can't shop online.
If you can list the street address of the Post Office along with the P.O. Box number the USPS will accept it.
The reason FedEx, UPS, DHL, TNT, etc "don't deliver" to P.O. Boxes is that they don't maintain the same database of "which post office has which P.O. Box for which zip code" that the USPS does. If they don't maintain this mapping, how is the delivery driver supposed to know where to go?
Post offices have very short hours. And most packages are delivered via UPS, Fedex etc. Their hours aren't much better and they have far fewer depots. When DHL still served the US I once had to drive 45 minutes (each way) to get to their depot which was only open 9-5.
If you have a car, going down to the local UPS or FedEx depot usually isn't too bad but if you don't have a car (e.g. most people in NYC) it can be a real nightmare. In urban areas, depots tend to be in industrial areas with poor mass transit access. Accordingly, having a local, secure, place to have packages dropped would be a great service for the latter group.