I've been a prime member for years and I just (naively, I guess) assumed they ship everything UPS, because that's the only way I have ever received a package shipped by Amazon.
I'm in a suburb and whether it's UPS, FedEx or USPS, my item is always left on my porch unless signature requirements were specified. In the nine years I've lived here, packages have never been stolen. I, however, would prefer if Amazon shipped via USPS. My mail is delivered, very consistently, between noon and 1:00 PM. For UPS, however, I am almost always the last delivery, which is at 7:00 PM and sometimes later.
I also found it interesting that this person received so many credits for screwed up shipping. I would expect a company like Amazon would have analytics in place to monitor the frequency of these problems and intelligently pick the best shipper. Reading the comments and the problems that others have had with FedEx and UPS makes me wonder if the issue isn't so much USPS vs. FedEx vs. UPS, but instead the location being shipped to isn't served very well by any of them and USPS was picked because they have more pick-up locations conveniently located. Getting the "slip" in my area from UPS means driving about 20 miles to stand in line versus the USPS, where it's only 2 miles. But that's just a guess.