If everyone expects to have health care insurance run the same way, then it can only reduce administrative costs and benefit consumers. A single-payer system across the board means there are no surprises with coverage and how medical care is paid for. Everyone wins in that there are more streamlined plans and consumers know what to expect.
Currently, doctors and their employees spend some portion of their time dealing with insurance companies to verify patient coverage and get approvals for treatment. A single-payer health care system would largely do away with this issue, saving time and money. Standardization of health care forms and having one rule about medical testing is done (versus different insurers requiring different things to verify that a patient has the condition that the doctor wants to treat) would save time and money. There is something like $20 billion annually to be saved just by sorting out issues like these, and they're automatically addressed with a switch to single-payer.