We pay the middlemen, yes. I don't see how we pay for other people's healthcare. The private insurance that I've experienced takes many factors into account (age, quality of health, pre-existing conditions and so on). Thankfully because I'm both young, and don't have pre-existing conditions, I pay less insurance premiums than a kid born with diabetes.
Remember, we're talking about technicality here. We aren't talking about ethics. Strictly from a money standpoint, we're not paying for other people's insurance.
Cheaper for everyone except higher income folk. They would benefit from a multi payer, private insurance system as they would end up paying less.
Insurance premiums aren't decided based on my income. They're decided based on the probability of me needing healthcare. Therefore, we kinda are not doing that right now. Universal, single payer healthcare would mean that healthcare expenditure would increase with my income. If I'm rich, I would be very sad.
But I'm not. Also, eat the rich. Healthcare is a human right. I am very happy with the universal healthcare that I have lol. I wouldn't want it to go away at all. But again, I was talking about the technicality here.