Having worked in non-profits extensively, it's also disingenuous to claim that non-profits answer to anyone other than their primary donors, whose priorities are frequently not aligned with regular people.
That's a shockingly high amount, especially if you don't get healthcare out of that.
16% of my taxes go to pay for my health care system, which covers the whole country. For someone on a median income that's 3% of their income. For someone on a top 10% income that's 4.15% of their income.
To spend 5% of your Gross Income on the NHS in the UK, you have to pay 31% in total taxes, that's an income of £75k, that's circa top 5% income.
2. Charity has had thousands of years to solve societal problems, yet it didn't. What you're describing is a pipe dream.
Within the span of a less than a century, countries around the world have addressed healthcare effectively such that care is available to most, if not all.
[1] https://en.wikipedia.org/wiki/Martin_Shkreli#Daraprim_price_...
2. Charities. You can start your own: give 5% of your income to underinsured. For example:
http://www.philanthropyroundtable.org/almanac/hall_of_fame/j...
At the urging of Frederick T. Gates, perhaps his most trusted philanthropic adviser, Rockefeller became increasingly devoted to medical research. In 1901, he funded the Rockefeller Medical Research Institute in New York City. Modeled on the Institut Pasteur in France and the Robert Koch Institute in Germany, it was the country’s first biomedical institute, soon on a par with its European models. The results were dramatic. Within a decade, it created a vaccine for cerebrospinal meningitis and had supported the work of America’s first winner of a Nobel Prize in medicine. Today, known as the Rockefeller University, it is one of the leading biomedical research centers in the world. Twenty-four Nobel Prize winners have served on its faculty.
But don't force charity on others who may be struggling.