It's interesting - in Australia we have universal healthcare. But it's not like this.
We have something called Medicare which covers hospital, GP and other medical expenses, it's free for those making under a certain amount per year.
We also have something called "Medicare rebate" for middle class people who go to see a GP, it usually covers half the cost of the appointment unless they are a GP that does "Bulk billing" which means that Medicare covers everything. Bulk billed GPs are always packed so you often have to wait for hours in the waiting room to see them.
Medicare covers mental health too, up to a point. If a GP gives you a "mental healthcare plan" you can get up to 20 Medicare-subsidised (or bulk billed) sessions with a psychologist per year.
If you make over a certain amount of money per year, when you're being taxed, you have to pay something callled "Medicare levy" which is an extra 1% or so on your tax. Unless you have private health insurance. If you have private health insurance, you don't pay the Medicare levy. That's an incentive for middle class and wealthy people to not overstretch the Medicare system and allow it to function well for those who need it more.
It's important to realise our private health insurance works differently here too. In Australia, private health insurance covers specialist doctors, optometrists, dental, physiotherapy and other forms of healthcare that aren't essential like acupuncture, massage etc. There are many providers, many that offer very cheap insurance. My insurer is AHM which offers a very cheap rate (about $6/wk) but gives me 2 free dental sessions per year, as well as up to $200 for glasses/year, optometry checkups, and 50% off physio and other therapies up to a certain amount. It also gives you about 50% off many specialist doctors' services. You can get more expensive packages if you need more discounts etc.
Even if you do have PHI, you don't need it for GPs. GPs are always covered by Medicare. And you don't need it for the ER in hospital either.
I broke my foot a few weeks ago and showed up at the ER at my local hospital. They gave me an Xray, pain medication, and a moon boot and crutches, and I didn't pay a cent! It was all covered by Medicare.
Finally, our pharmacies do not charge based on Medicare or PHI. We have something called Pharmaceutical Benefits Scheme (PBS) and Pharmaceutical Safety Net which means medication over a certain amount per year becomes free. So cancer patients for example do not become bankrupt for chemo. And we don't charge insane prices for insulin either.
Private health insurance does not stipulate what brand medicine to take either. You can almost always choose a generic brand for about half the regular price.
It is possible to have universal healthcare in a way that doesn't compromise on quality of care, and I think Australia does achieve that.
That said, Australians on average pay a much higher tax than the US. I'm making about $120k/year and pay about 32% tax. We have good salaries and a high minimum/award wage though so it's not a major hardship.