Nothing in life is ever free! The correct term with the NHS is "free at point of use". So as you know, it LOOKS free when you need it. Just saying that to avoid Americans commenting on it.
What they have in the USA is no system at all. Imagine a Britain with no NHS, just doctors and hospitals where you have to pay every time you use them. This gets expensive (it's the most expensive health care in the world because doctors charge so much) so you want to have insurance. Poor and old people get paid for by Medicare and Medicaid, taxes pay for those, and the "Obamacare" law has made it compulsory for everybody else to have health insurance.
So one big difference is that in the UK, taxes pay for your health care unless you choose to "go private", but in the USA, unless you are entitled to Medicare or Medicaid,,you have to buy insurance. If taxes look higher in the UK, which they aren't particularly, it's because they pay for more!
So you have insurance, and that will pay your medical bills. To keep it simple, let's forget things like excesses and exclusions in the insurance policy. Insurance pays the bills, but these can be big bills and so the insurance costs a lot too.
The real issue is the amount American doctors charge. They charge a LOT and this makes Medicare, Medicaid and insurance expensive. The enormous difference with the NHS is that the NHS itself employs and pays the doctors and runs everything, so it can control what doctors get paid. Doctors can work privately like all American doctors do, but they won't make a living unless they have a lot of rich patients willing to pay for it. The NHS pays a good wage to doctors, but it won't make them millionaires.
And that is the biggest cost. If you can control what you pay doctors, you can have an NHS to cover everyone.
Now here's a bit of economics. Chemists and hospitals buy drugs from drug companies. One individual customer can't bargain much. But if the whole NHS says "we want 1 billion Ramiprils, can we do a deal?", you bet they can come up with a discount! We have a monopoly buyer situation. If the company says no, who else are they going to sell to? So that keeps costs down too. (I mention Ramipril because I have to take it every day for high blood pressure, lots of older people have that, so I'm sure the NHS actually does need billions of them.)
Your Dad is wrong. Of course doctors will treat you if you're sick. Any doctor is sworn to preserve life and that's why American doctors won't do lethal injections. The problem comes with who's going to pay for it. In the UK, the NHS just pays. In the USA, you're reliant on doctors doing some charity work. This reminds me of my Mum, who was born before the NHS started in a charity nursing home.
The NHS is a very different way of running things. And a very good one too. StephenWeinstein is right - if the USA actually had an NHS, it could probably do it just for the amount it already spends on Medicare and Medicaid, But it won't do that because all the doctors and health insurance companies will scream, and they bribe the politicians not to do it.