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Hell, at the moment,health care threatens to grow to consume the entire economy of not just the United States, but most of the developed world. If you want to use your go to example of Europe, they too face rising health care costs. Check our "Wise Medical Insurance" to read articles on how to save money on health insurance.

I never heard an error defined as a small mistake. An error can be huge, eg the Church excommunicated him because his beliefs were in error.
In teaching a language, instructors look for the difference between error & mistake. A mistake is made through inattention or slip of speech, and the student will correct immediately when s/he sees the mistake. But an error is the result of an incorrect belief and is consistent. The student needs to relearn a grammatical rule or pronunciation to correct an error.
This is the same for computer programming: a computer makes an error because it has been programmed to consistently respond in the same incorrect way to a prompt. A computer does not make a mistake, but a computer programmer may make one.

You mentioned doctors acknowledging mistakes: My own birth was very difficult for my mother and for me. Afterward, the doctor who operated on my mother (it was a C-section) sat down with her and said that he was sorry, that the labor should never have been handled the way it was and I would not begrudge you at all if you sued for malpractice. My mother didn't. In fact, today, almost 37 years later, she still remembers that doctor fondly for owning up to it.

It's not that "rogue doctors" cut off the wrong leg! That kind of thing happens because patient records get mixed up or it isn't indicated clearly enough which leg is supposed to be amputated.

my first marriage was a great mistake, and i say that without sarcasm. were it not for those years of difficult and very often ugly emotional work i wouldn't be capable of the healthy relationship i have now. embrace the imperfect; turn and learn!