We pay for lots of services that only other people end up using. It's part of being in a society.
When something is reasonably considered optional, ideally and hopefully you aren't forced to pay for other people's partaking of that optional service.
Healthcare is not one of those things. It's optional in the sense that everyone has the option of refusing medical care, even if sometimes that means dying, but it's not optional in that almost every sane person with more than a minor cut or sniffles or food poisoning opts for medical care if it is available rather than the alternative.
Other mandatory goods and services, like food, clothing, housing, city utilities (water/electricity), even internet access, work better in a mostly private model for a couple major reasons. First, the price variance for the necessary part of those goods and services is very low compared to variance for the medical costs someone might incur. Second, a lot of people can afford and want more than the bare necessities. Not so with medical care, where few people can afford major surgery or routine treatments for some major chronic conditions.
Another aspect of medical care is that we benefit even if we're never treated. Do you know anyone who has ever needed significant medical care at a hospital? I doubt your relationship with those people means nothing to you.