The parents income doesn't just magically create wealthy kids: money runs out. In the US, wealth controls the child's access to good education, and parental wealth may also result from good genetics (intelligence) or culture around wealth and investment.
An example: friend of mine isn't particularly intelligent, but she's wealthy because she knew the law of 72 and said no when offered a credit card (with compound interest like all credit cards) before her frontal cortex developed. Most young adults would say yes and immediately enter financial slavery.
Parental income is the highest predictor of a person's income. Nothing you have said counters this argument.
If you have any stats or data that counter it, then please provide them.
It would be entirely possible for both to happen: an actual competitive health care market in the US reducing costs to closer to the rest of the first world and the government paying those costs via taxes (pretty close to what people in the US pay now).
Let's take a look at Bill Gates' luck. He was at a private high school which was only of a few in the world that had access to super-ish computer. Not only that, he was given control access to do whatever he wanted.
Sure, he was talented, but if he hadn't been at that EXACT high school at that EXACT time, he wouldn't be a billionaire.
Billionaires need a shit of ton of things to go right and ALL their decisions need to work out almost perfectly.
Even when they "lose" something, if you became a billionaire, you found a much better opportunity right after you "lost" something.
The fact is the number one predictor of a person's income, health and net worth is their parents income. How do you explain that the extremely high correlation of parental income and children income?