I don't see how Amazon can do it cheaper (cost-wise)/better than UPS
One word: Teamsters.
UPS drivers are Teamsters. Which means they have good paying jobs with pensions and fringe benefits.
All Amazon has to do to be cheaper is to hire no-benefit minimum wage part-timers. Or even better, just use independent contractors. It's the Uberization of package delivery!
It will be far cheaper, but it won't be better.