Medicare focuses on the elderly, it's not really a great metric for how much healthcare should cost. The VA system also swings towards the elderly, and also supports a minority much, much, more likely to be suffering serious health problems than average. Which leaves Medicaid, which has multiple levels of bureaucracy (often corrupt) in addition to the normal levels you'd expect because it's a combination federal and state program.
There's no real reason to suppose single payer would cost more than regular private insurance, and a lot of good reasons to suppose it would cut costs. The amount of absurd bureaucratic hell inherent in private insurance is staggering, which is duplicated across multiple insurance companies. Healthcare costs in the US for alike treatment is staggeringly out of whack with the rest of the world.
It also gives government incentives to promote other solutions that would help. We need way more doctors, our system is fundamentally under capacity, which is why it has to be rationed using copays. The government has no incentive to fix that because it's not its problem. The healthcare industry has no incentive to fix it because doctors like being paid more money. The insurance industry has no incentive to fix it because it applies to all insurance companies, so no insurance company gains an advantage by starting programs to train more doctors. You can only fix this by giving one entity an incentive to fix it.
Obviously, despite a single payer system costing less than a multiple insurance for-profit system, it'll require tax rises, but if those tax rises for the vast majority of people are lower than the cost of their insurance premiums, it'll obviously balance out. That is a point advocates need to hammer home though.