That might be how a delivery service works, but it's not how the post office works in almost all the countries in the world. Including, I suspect, the USA.
Post office networks are actually run as networks. They have drivers who collect from post boxes at regular times, sorting offices that work round the clock, postmen who do regular delivery rounds, and so on. The cost of running the network is, over the short term, almost impervious to the amount of traffic.
(Over the long term, of course, you restructure the network to cater for things like population changes -- again, just like the Internet.)
What you should realize is that a lot of people used to think like you before 1840, which was why the Penny Post was such a big deal. Network thinking made mail dramatically cheaper and led to a massive expansion in communications, and the whole system went global in a very short time.
None of that would have happened under the old system.