I often try to go cash pay, since in many cases at this point it's actually cheaper, absurdly enough, and get told that I can't even get an appointment that way. I actually got kicked off primary care at a regional (fairly large) practice because they couldn't take my insurance, would not bill out of network, didn't service uninsured people (!!), and would not take cash, credit, barter or trade.
The advantage of the less developed countries is that you can get pretty good private medical care for a reasonably price. The quality may not match the very best care available in the rich countries, but in many cases, it is adequate.
What is a "reasonable price"? Relative to a rich country, the price may be low. But relative to the average income to common folks in that country it is likely just as expensive as a comparable private option is in a rich country. Apples vs. oranges.
Iow, you are describing the perspective of a rich person in a poor country. I don't think a rich person in the U.S. (top 0.1 percentile, let's say) minds the health case costs in the U.S. either.
Of course the top tier private hospital and specialists can be pretty expensive, but is not that expensive to avoid the public/government health system.