Mr-A, Quite take the point that big cities are a separate case, but people in the country are going somewhere to buy their food and so forth, often fairly frequently. A bit of creative thinking could get their PO boxes into pubs, shops or wherever they're going anyway. Also, the revenue from a post office ont he current model relies on people actually wanting to post things doesn't it? Whereas a PO box rental model relies on people renting a box in the hope of receiving things, so the income is perhaps more predictable and stable.
It would take some getting used to, but I suspect that a lot of the places where home delivery isn't done are places with much worse communications links than much of rural Britain.