False dilemma fallacy:
a type of informal fallacy that involves a situation in which only two alternatives are considered, when in fact there is at least one additional option. The options may be a position that is between the two extremes (such as when there are shades of grey) or may be a completely different alternative.
Your premise is that there are just two options: A, people are refused treatment or B, we
force others to pay for their treatment.
It's already been pointed out that free clinics exist, doctors do pro bono work and there are charitable funds that exist to address the issue as well. All of those are voluntary exchanges and do not require the use of force to reach the desired outcome - i.e. people who can't afford it still get medical assistance.
If you feel it's immoral to deny people medical treatment because they can't afford it, you are free to voluntarily contribute money to whatever clinics or charities exist to cover the costs. However, you do not have the Right to violate the rights of someone else by forcing them to contribute to a cause that you believe is moral.
Cheaper for whom? Here's a "real world" example from my life: My full time job offers HCI, it's a group policy for all the company's full time employees. My employer covered 30% of the cost leaving me with a monthly premium of $180, that policy was costing roughly $255 a month. With the exact same HDHC policy but as an individual, rather than part of the group, and from the exact same insurance company, my monthly premiums are $96 a month.
The extra money I was paying on the group plan was to subsidize the HC coverage of people who were older and in poorer health than myself. The healthy people in the group pay more than they have to in order for the less healthy people of the group to pay less than what they should.
Broken window fallacy –
an argument which disregards lost opportunity costs associated with destroying property of others, or other ways of externalizing costs onto others.
So when you say it's "cheaper" to have everyone in the same pool, that only benefits the people of poorer health and that benefit comes at the expense of those who are in better health. The cost of providing HC to any specific individual doesn't actually change, so you haven't actually reduced the cost of providing HC to anyone. Collectivization doesn't reduce the cost of a single band-aid, much less reduce the cost of purchasing or operating an MRI machine. All you've done is spread out the individual costs among the entire group - making it cheaper for some but more expensive for everyone else.