Within the NHS specifically, there was the recent case of Alfie Evans[0], where the medical practitioners said there was no hope for him and the family wanted to take him abroad for care. The NHS and the UK legal system decided that his best interests were to die in the UK. His family was not allowed to take him for additional care.
That isn't really an issue with the NHS. If it's an issue it's with the UK's family law which recognises that the parents of a child doesn't always act in the child's best interest and allows the state to intervene.
There are better systems... over here in Uruguay, you have basic tax-paid "free" healthcare but you are able to purchase better services on top (I have several layers of extra care on top of national basic service).
The richest usually purchase insurance to be operated in the U.S. by top doctors (ironically, probably cheaper than what U.S. nationals pay for the same service), but local doctors are very good (there ARE some services where U.S. expertise is preferable).
The issues you mention are not directly health care related. There have been analysis of the mortality rate in the US, and when you remove people who die from things like being murdered or in automobile accidents and dying without receiving care, the US has the best mortality rate among developed nations. The data is old, but there's no reason to think it has changed dramatically in the last decade.
Obesity rates and chronic diseases that are the result of lifestyle choices are not health care issues, they are public health issues or public health policy issues. No doctor can legitimately be assigned responsibility for citizens who live unhealthy lifestyles.
When comparing cancer survival rates the US is, again, among the top for many types of cancers. It's almost impossible to compare infant mortality rates because the US doesn't adhere to any international norm for what is considered a live birth and reviews of the data have shown that the US will frequently label an extremely low birthweight child as a live birth, while many other countries will label the birth as a still birth.
The reality is that actual care in the US is what you'd expect from a developed country. The reports that put the US down the list of first world health care systems universally factor in cost and accessibility. These are real issues, but it's not the same thing as "paying more for crappy outcomes."
Countries with different systems have their own issues, which is why I pointed out situations like Alfie Evans. There was also Oliver Evans whom the NHS said could not be saved and would not get treatment. The family paid $170,000 out of pocket and crowd funded another $100,000 to bring him to the US where surgeons in Boston successfully removed a non-cancerous tumor from his heart and saved his life.