Considering Republicans have been trying to kill the USPS for a hundred years with increasingly restrictive policy changes and increasingly impossibly tight budgets (considering the scope of their task with heavy competition), I think you are entirely mistaken about where the problems are. That the USPS operates at all under such absurd restraints and tight budget is nothing short of miraculous. The problem lies elsewhere, namely, with the major political party that would strongly prefer no government services, and no government, really. Other areas of the world call them anarchists; here we call them Republicans.
Narrowing down the USPS debate to "operate" vs "not operate" is a false dichotomy. For example, how critical is 6-day delivery schedule to the least populated areas? Would a 5-day schedule work out reasonably well?
It could change things like Frequency of delivery, or other cost cutting but I have advocated for that anyway even for non-rural routes. personally i think residences should get mail delivery on Tue and Thur, and businesses on Mon, Wed and Fri with none of weekends. that alone would be a huge cost saving.
Idk, I think that a USPS run like a business would either just decide not to operate in the less population dense parts of the US or charge them exorbitant rates for service. Even as someone who lives in a dense urban area, I think it's generally a good idea to have guaranteed mail service to anyone in the country even if that means that I'll essentially be subsidizing the cost of delivering mail to rural areas. I have trouble believing that USPS could be profitable, affordable to everyone, and available to everyone, and I'm much more willing to sacrifice the first one than either of the other two.