If the campaign to eliminate driving from most people's lives is successful (whether through autonomous vehicles or public transit), I expect activists will move on to the next most dangerous activity. Something will always be the leading cause of preventable injury/death and certain people will always be enthusiastic about banning whatever it is.
I think involving prevalence in the equation is ethically inconsistent. A society that bans drugs on the grounds that they incur a medical cost and provide limited utility should clearly ban base jumping as well.
Search and rescue is just one element of saving somebody who has hurt themselves climbing or base jumping. If I understand the policy GP described correctly, if the participant were poor enough, their physical therapy might be covered. Certainly surgery would be.
Differences between how we treat adrenaline junkies and drug users today are really just details. The broader moral question remains: Does a society that has committed to providing medical assistance to those who cannot afford it not tend towards limiting behaviors that tend to cause unusually high medical bills?
These activities are hazardous. In a scenario where the government manages health hazards as a consequence of managing health, government regulation of ice climbing or paragliding is logical.
Again, this was hypothetical, not extant.
Society must pay for drug rehabilitation -> society should consider banning drugs.
Society must pay for medical bills -> society should consider limiting behaviors that have a high probability of incurring medical expenses.
These are the downsides to welfare programs that people should really be focusing on. It's not the cost in terms of dollars, but in freedoms that it calls into question.