I assume you are asserting this as a consistent principle and not just a one-off. IOW, you also would oppose spending tax money on other kinds of self-inflicted damage, like smoking-related illness, cirrhosis of the liver due to drinking, diseases associated with obesity, preventable diseases like AIDS, etc.
Land use and any associated fees are collected by the Mt. Hood National Forest since it is federal land. Search and rescue on the mountain is managed by the county. What needs to be determined if any of the use fees (federal income) are diverted in part to the county to offset any SAR operations (county expense). I doubt it.
I have the solution, let them climb only after they post either insurance or a large bond they would get back if not used for a rescue. Everyone would still be free, yet made to be responsible.
The insurance regulators in that state would require rescue insurance to be offered if the company sells other kinds in that state, simple easy, and elegant solution to the whole problem. It is really no harder to do than requiring hunting/fishing licenses, and game wardens would arrest anyone without the coverage with mandatory jail time, climbers would get the coverage.
Why not have everyone put up a bond against any kind of activity that could require a large amount of rescue effort, such as driving a car? Where does society draw the line, cause I would certainly not draw it here.