How would you feel if you had kids and one of your kids took up base jumping? What if they took up high school football?
Once you've carved out exceptions, you've now changed the frame to something horrible. Systemic problems (obesity, drug use, mental health) probably deserve government meddling, but your first sentence is exactly the problem with conservative points of view. Conservatives can't put themselves in the position of doing something reckless and therefore feel they deserve to be better off (more money in their pocket) than the other person, until something bad happens to them. "Fuck you I got mine" is the liberal characterization of this attitude.
I was hospitalised for a few days because my cat bit me - is keeping a cat reckless behaviour?
I think it would be supremely silly, and more importantly hugely expensive, to try and monitor behaviour and track down what people were doing that caused an injury in order to decide whether their health care costs should be covered.
Like, if you want to go, say, base jumping (just picking a risky recreational activity here), I'd be OK with saying "you need a separate insurance for that, if you hurt yourself the government isn't picking up your tab." That seems like a more straightforward way of handling the problem than to try to somehow tax base jumping.
The trouble is, of course, where to draw the line. It's possible that the cost of these voluntary risks is just negligible compared to general health care.