Indeed. We already effectively have single payer system for large segments of the population, the Medicare itself. It still didn't push the costs down to European levels for Medicare recipients.
The US needs cheaper healthcare, but single payer won't achieve that.
Part of the reason for that is that Medicare is forbidden by law to negotiate prices the way other insurance companies can as a result of the GOP's Medicare Prescription Drug, Improvement, and Modernization Act of 2003.
The efficiency gains of single payer are overrated. We pay more than other countries do in admin overhead, but admin overhead itself is a small fraction of total costs, and as a percentage of our spend, admin costs would necessarily rise, probably dramatically, if Medicare starts covering 30 year olds in addition to 68 year olds.
† I'm just taking your number here, not interrogating it.