Archie I get what you're saying and you make some valid points. However, reality is a little more complex.
Medicaid and Medicare payment rates are set by law -- the hospital or physician has no say in the matter. Medicaid payment rates vary by state but even in the highest paying states hospitals don't even cover their costs -- they lose money on every Medicaid patient that comes in the door*. Medicare payment rates are better but they may just barely cover costs, or may just fall short.
Payments from commercial insurers are where providers make their money. They will negotiate deals with all, or some, of the health plans available in the market. They generally give better deals to the larger plans who represent a bigger portion of their patient base. This makes sense from the point of view of them running their own business, but it has adverse macroeconomic consequences. It gives the largest health plans a built-in competitive advantage over smaller ones: lower payments to hospitals and doctors means lower premium rates.
As for the uninsured, what you say is technically true but again here the reality is muddier. A few states have laws that require providers to accept Medicaid or Medicare payment rates. In other areas, a hospital system will usually provide some sort of reasonable discount off of billed charges to uninsured patients that is consistent with the discounts they negotiate with mid-level commercial carriers. (Doctors operate a little differently, and for routine office visits will often not provide discounts).
Hospital billed charges are generally only paid by visiting Saudi sheikhs and the like.
* - technical caveat on this statement: measuring of hospital and physician "cost" is a gray area with lots of room for judgment. When I say most hospitals lose money on Medicaid, I mean on a fully-allocated fixed cost basis. On a pure marginal cost basis, they are probably at least breaking even.