I'd rather have a bureaucrat I can vote out of office
Those aren't the type of bureaucrats that would be running the system.
Ultimately, they would be more accountable to the people.
But let's look at the issue here.. it is often argued that government provided healthcare increases bureaucracy and wait times, and yet here is a great example of private health insurance companies deciding that someone no longer has asthma despite never having been in contact with the doctor about it.
If anything, we need to remove all incentives and benefits to doctors provided by drug companies to prescribe new, expensive meds. Then we need to remove the ability for insurance companies to question a prescription written by a doctor if those types of prescriptions are covered under the policy.
The doctor should have the final say on what treatments are used. If the insurance company thinks a doctor is being unreasonable in general, they can drop them from their network.