You have mistaken the role of government in healthcare. Government, by mandate of the people, already requires treatment of acutely ill individuals, and nobody is arguing to change that. The question is then how this is paid for (mandated insurance or socialized medicine), what is more cost-effective (in terms of preventative care), and is earlier treatment sufficiently more humane to be preferable in some cases? Also your "small minority of the population" is simply not small. A good 30-50% of Americans have trouble affording health care, partially because people like you prevent the appropriate management of the cost for the already existing basic universal health care. It's not all or nothing as you imply, a typical successful model is to provide basic coverage by mandate or required insurance, and then individuals with means can pay for other perks like shorter wait times or better rooms.
On the dental care side. If society places value on not having a bunch of snaggle-toothed people around, regardless of their ability to pay for orthodontic care, then the government can similarly intervene in that market on behalf of the public to make it more affordable.