The problem at the very outset with this argument is that a system where patient pays hides the queues, because
everyone can afford to go.
As such you can not directly compare a system where going to the doctor costs money with one where it does not.
This is one of the biggest problems with the US system: Nobody knows the extent of the unserved need, and what that ends up costing society in lost productivity (never mind the human suffering) because people avoid healthcare services they know they can't afford.
But we can compare Canada to other socialized systems, and many do more with less and still end up with shorter wait times.
The vast majority of the worlds healthcare systems also do not prevent people from going private.
Most places also allow you to buy your way past the queues. As such, the difference is stark: Most places you have the choice of waiting, and paying much less, or paying for private services and get seen as quickly as those who can afford it do in the US while a large number of people in the US has no access to a lot of healthcare services at all.
Want to bet whether the people who can't afford to buy healthcare services would prefer waiting 20 weeks - or any amount of weeks - to get treated vs. getting no treatment at all?
Now consider that average US spending per taxpayer in the US on Medicare and Medicaid is substantially higher than the per taxpayer spending to fully fund universal healthcare in the UK for example. And then on top of that Americans buy private insurance since most are not covered by Medicare and Medicaid.
Most Americans pay twice. Meanwhile in the UK only 10% find it worthwhile "paying twice" by having private insurance, even though most private insurances in the UK are very cheap for the simple reason that there's simply no demand for replacing A&E, ICU type services, and relatively little demand for covering private GP services (most people can be seen by their GP quickly, and a private consultation on occasion is cheap), so the insurances mostly cover seeing a specialist quickly if the NHS has a waiting list, or getting surgery quicker if the NHS has a waiting list.
This is what I really don't get about the US debate: The poor regulatory regime basically mean Americans are already paying enough towards healthcare to provide universal coverage via taxes alone if the regulations were properly reformed. A start would simply be to remodel Medicare after e.g. the NHS model, and allow it to exercise its market power to drive down costs, and use the savings to extend coverage.
As it stands, the inadequacy of the US healthcare system is largely down to corporate welfare in the form of regulations that actively prevent the government from providing cost-effective healthcare services.