With Priority Mail now offering $50 in free insurance to the flat rate packages, it's now the obvious choice for my online retail business.
Even before then it was the clear winner. Flat rate standardized (free) packaging is such an awesome perk. USPS loses way too many of my parcels, but it's just not worth going to UPS/FedEx and paying 2x the rate plus way more hassle for a slightly reduced lost package rate.
And I'm the opposite. I'll pick USPS over UPS/FedEx every time. In over 20 years of shipping I've never had a package show up at my doorstep damaged by the Postal Service. UPS can't even ship me their own envelope without ripping it. Don't even get me started on when amazon uses Lazership...
Of course you know what they say about the plural of anecdotes.
It is good practice to offer multiple shipping options. Package delivery differs greatly by location. In some areas, the postal service is the best. In some areas, UPS. In some areas, Fedex.
That having been said -- because of the discounts offered to large shippers, the non-preferred carrier may cost 2x or 3x as much. For example, when Newegg switched from Fedex to UPS, they continued to offer Fedex. But they passed along the difference in price.
The number of customers I would gain by making that available doesn't outweigh the hassle of requiring another shipping provider. It's not even close (I've tested it).
When I was reselling used books, USPS was the best ROI for shipping (packages and media).