Unfortunately it would also make people think twice before taking their kids for a walk in the Lake District in good conditions. We should think very carefully before labelling our national parks as places where you need insurance to step off the tarmac.
As others have said, it would create an unwanted bureaucracy around the charging, chasing and division of the funds.
I've not noticed that happening in respect of Ambulance trips for motorcyclists and other "self inflicted" problems, where that has been happening since the 70's. Nor is it noticable in the Alps where helicopter rescues are charged as a norm. In essence its the same argument about the Royals being a toursit attration. i.e.not a shred of evidence to support the supposition, but if you say it loud and often enough people start to believe it. That apart, in what way would making people more cautious of the consequences of say, going up Snowden with inappropriate footware/kit/food/preparation be a bad thing?