Single payer can mean a variety of systems.
In Austria and Germany, the insurance covered prescriptions, most dental care, just about everything. We paid a small, very small co-payment for prescriptions. I don't know if that is still the system.
Single-payer in my experience means that your insurance premium is taken directly out of your paycheck before you receive the check. Each person pays a share depending on his/her ability to pay. The whole family is covered. Everyone is covered.
Government control? What is covered is determined by the government. In most countries, you can buy additional, private insurance if you, for example, want a private room or something not covered by the government plan. Some countries use non-profit insurance companies. Some have direct, government insurance. It varies.
Your relationship is with your doctor. Your doctor gets paid by the government.
Virtually all doctors were on the single payer plan so finding a doctor you liked was easy for us.
I liked single payer.
The people who are not covered by health insurance under Obamacare are often those who need coverage the most. I like Obamacare but it leaves a lot of people without insurance and adequate medical care.
Single payer is a little like Medicare. You don't have to worry about whether you can afford to pay for medical insurance. But in Europe, in our experience, the co-pays were much lower than with Medicare or American insurance. I don't know whether that is still the case.
But single payer means generally that the money for your insurance is taken out of your paycheck and assessed according to ability to pay -- and there are not for-profit insurance companies.