Plenty of retail stores offer shipping to remote locations, so a hike in postal rates would either hurt them or push them to UPS, etc.
Also, it's not the US Postal Service's job to prop up local economies.
Why is Amazon successful?
It's not just cheap shipping; it's a wide selection. There are folks who need gluten-free foods, for example, and they can't find them in their small towns. Or they want to discreetly order LGBTQ literature or they want home health appliances or products (e. g. Depends or the like) and want to maintain their privacy. Amazon allows for that.
Amazon can also maintain cheaper prices for a lot of things (and rather expensive prices for others; it pays to check costs per unit) because they don't need to store too much stuff. Local stores need to stock multiple copies of items on their shelves so things don't look too bare. A big part of the American shopping experience in particular is a show of abundance. So they need 20 packages of toddler diapers, etc., whereas Amazon can technically just have one in stock.
Why is the US Postal Service failing?
Being run by the government means their budgets are slashed in ways that UPS's and DHL's aren't, and there are political pressures of one kind or another to save or to move money around. The Post Office also kind of let UPS, etc. eat their lunch, by offering same-day service and other amenities which the USPS could not afford to do for quite a while. At the house where I grew up, they used to take the mail, but that was because it was an exurban kinda area and the mail truck driver would just deliver and then take at the same time. I have never lived anywhere else (in Boston, Wilmington, Philadelphia, or Providence) where that was done, and my folks moved one town over and they lost that service, too. I expect it won't survive much longer in my old home town, and home boxes might have to be switched to mailboxes either at the local post office itself or some sort of apartment-style box of boxes in one location. Why? Because there are massive hills there, and it's a pain when it snows. If the letter carrier doesn't have to make those hills, then their trip shortens and it's safer for them (hence maybe liability insurance rates for them go down).