The US system is worse on every single metric (unless you're rich of course - which is the point of it).
Source: someone who is willing and able to pay for services, but cannot find them
If you took half of that difference and applied it judicously, waiting lists would drop quite a bit.
I have Medicaid in the US and I get seen within a week to a month (sometimes even same-day) in non-emergency situations, and 100% of my medical bills are covered by the government.
The worst part of it is that for certain specialties there just aren't many doctors who accept Medicaid, so I pretty much have to take what I can get... but this shouldn't be a problem if every doctor accepted a single-payer plan.
On the other hand, for things like mental health care there's a large amount of choice in my area. So overall I'm pretty pleased.
The situation would not be the same if everyone was on the same payment plan.