Of course they are not unavoidable, they are just
a lot likely to happen when you go skiing than when you don't.
The problem with socializing risk is that you get the tragedy of the commons when some people choose to take high risks (skiing is definitely in the moderate end of "extreme sports") and pass off the risk to the "commons".
In Denmark, where health care is socialized, the argument that "society is justified in regulating activity X because we will pay for your hospital stay if/when it goes wrong" is absolutely a part of political discourse. Unhealthy foods, not exercising enough, smoking, mandatory bicycle helmets etc., it's pretty universal.