"by comparison the us system is completely more affordable since most is paid by private money"

Let me help you understand this.

For example, in US: You make $50K, then your employer pays $10K in benefits and you out of pocket co-pay to provide health care for your family. If you lose job, you have an option to continue paying $10K. If you can't afford that, you lose coverage. This means that doctors and hospitals will turn you away for anything but life-threatening conditions. This means that if you get critically sick and have to go to the doctor, they go for your assets, like house, to recover these costs.

Prices for medical care in US are much higher. In Canada, provinces dictate prices. This means MRI costs are known upfront (and are not paid by you). In US prices negotiated with medical insurance providers who get steep discounts on volume. This means that individuals when directly billed pay much higher price. This means that you don't know what your bill is until checkout.