The most common failure mode for single-payer is scarcity. You won't be denied, but you'll have to wait many months for an appointment.
> Would this not be replaced with faceless automata in government?
They are controlled by politicians, who are directly responsible to their electorate. Brexit is a good example, using money sent to EU for NHS was one of the more influential ads. Of course, the outcome turned out to be... different.
With the current insurance system, you don't have ANY levers. You can't usually change your insurance company because it's provided by your employer. And even if you want to buy medical insurance yourself via the ACA, you can switch it only once a year. With no way to tell in advance if your new company is going to cover your treatment.
You also can't even sue your insurance company if it denies you the treatment because _all_ insurance companies require binding arbitration. And arbitrators basically always side with the insurance company, because your contract says that the insurance company is always right. That's how UnitedHealthcare can get away with just randomly denying treatment.