Now I see their trucks nearly as often as Fed Ex and UPS.
The issues that Amazon has seen, particularly during holiday peaks is that shippers do not always want to continue to lower their rates and that shippers do not always scale up their operations to meet Amazon's needs during the holidays. For example: https://www.latimes.com/business/la-fi-christmas-shipping-de... and https://www.washingtonpost.com/business/economy/amazon-ups-o...
In response to shippers not doing what they want and to provide leverage, Amazon started to do their own shipping. This originally started as same-day delivery experiments in large cities as part of Amazon Fresh (grocery delivery service). It grew into non-grocery deliveries as a competitive advantage for Prime same-day deliveries and expanded from there.
I expect that "Transportation By Amazon" (TBA) where Amazon does package delivery as a competitor to UPS and FedEx will probably start in the next year or two, barring any anti-trust or legal problems.
https://www.buzzfeednews.com/article/carolineodonovan/amazon...
Your trusted Amazon delivery guy could fill your fridge while you supervise him through Amazon branded devices.
The program you're thinking of is likely "Amazon Flex" which is just part of their last-mile delivery system and uses independent contractors.
That said that they are /still/ smaller than Wallmart yet have scale sufficient to be competitive to mass shippers is amazing even considering the internet related physical letter shrinkage.