If you are going to permit "increased healthcare cost to a taxpayer-funded health system" as a line of reasoning, then you allow for a lot of absurd outcomes, like the government could forbid risky sports. This is not an "appeal to extremes" or a strawman argument, it's actually a valid reductio ad absurdum.

Opponents of public health care in the U.S. use precisely this as an argument: They say, if we're going to have taxpayer-funded health care, it gives the government a reason to start regulating aspects of our lives which we don't want any government to regulate.

There are only two ways out of that that I can see: The first is to agree with them and say: If we want to be a free people, we can't have taxpayer-funded healthcare.

The second (which I personally subscribe to), is to say that there's some kind of a "bar" that has to be met, and that "increased healthcare costs to the taxpayer" doesn't meet the bar of how much your freedoms need to be impeded by my actions, before it starts to justify government taking away my freedoms.