The problem with this line of attack on UHC is that there are probably just as many instances of bureaucratic bumbling at the hands of private insurers as there are at the hands of public sector bureaucrats.
Then what?
There is a lot of bungling in life. The difference is that a government agency generally can't be held accountable. When an insurer refuses to pay for a necessary procedure, there are government agencies, lawyers and the market to hold them to account. When a doctor earns a living from patients who trust him and choose him, he is motivated to do the right thing. When a doctor gets a paycheck from the government, he's going to get that paycheck no matter what. Poor health care is the result.
A private insurer will certainly try to save a buck, and it may take concerted efforts by patients, advocacy groups, the legal system and appropriate government agencies to overcome that. But when the entrenched bureaucracy says "no" where do you go?
The fact is, the lady in this situation was doing nothing wrong. She was using the government system everyone else in the country was at government (taxpayer) expense. But then she decided that the system wasn't working. So she took her own money (and some provided by others voluntarily) and sought to go beyond that. The bureaucracy would not stand for it. To whom does she appeal? Other bureaucrats? National leaders? Those who created, maintain and get paid by the national system? Where is the disinterested oversight?
There is a metaphor here that refers to a fox and a henhouse.