Now turn the table, you are the one with cancer. There is an expensive treatment that might or might not work, but it is the only hope left. How much is this wroth to you?
With money. It's like insurance, except paid through an extra tax.
> Or more accurately, why.
I mean, are you really asking why someone sees human life to be roughly of equal value, regardless of said human's wealth? But let's look at it differently: I don't think public healthcare precludes people from access to private treatments and additional private medical insurance. Public healthcare is just another baseline service offered by the government, which happens to remove the need for most of the people to seek that service in private sector. Like many other government services, you could view it as a formalization of helping others in need.
> There is an expensive treatment that might or might not work, but it is the only hope left. How much is this wroth to you?
Obviously, all the wealth I can command at the moment - meaning what I have, + what debt I can get into. Which is precisely why healthcare needs to be regulated, and why public healthcare is useful - private providers know that true demand price for many medical issues is essentially "all that you have", and they will try to get all that money. Inserting the government between the healthcare provider and the patient means the provider doesn't get to "negotiate" with an individual under duress.