Inisfallen notes that people have anecdotal evidence, and I would further that by saying that individuals experiences with a health care system doesn’t really paint an overall picture of the health care system as a whole.
California alone, one American state, has a higher population than Australia. Perhaps you have to take population size into consideration. What may be seen as successful in a country with a certain population size may not work in a country which has a population size which is 10 times bigger.
I don’t think there is any statistical evidence that any country with single payer has overall shorter waiting times than America does, perhaps they do for some procedures, but not when polled generally. Although of course it depends whether you have medicare or insurance but having short as possible waiting times is really important.