Please explain how a handful of delivery vehicles moving through a neighborhood each day for deliveries is more traffic than every single homeowner having 2-3 cars all of which are in circulation on those same residential roads each day.

The trucks are not a replacement for ordinary traffic.  They are in addition to ordinary traffic.  

Moreover, residential streets were not designed for trucks.  They are designed for cars.  The turns are often so tight trucks can't make them without crossing over into the wrong lane or climbing the curb.  The asphalt and subbase are thinner, and don't bear up under the load as well.  A parked truck on a residential street can bring traffic to standstill; a car wouldn't be parked there, or would be easy to drive around.    

By your argument, it would be better for everyone to drive to the post office to pick up their mail rather than having the post office deliver it.

I think we may be going back to that.  That is how it was for me, growing up in a small town.  UPS wouldn't deliver; too small a market.  The USPS would deliver, but only to the post office.  People had PO boxes and picked up their mail in town.  It didn't take as much gas you might think.  Most people didn't check their mail every day.  And everyone knew everyone, so you could have a neighbor pick your mail up if they were going into town.  Many people had their mail delivered to their office PO Box.  A secretary would go down to the post office daily, and pick up everyone's mail.