The government doesn't have this kind of search and rescue team because it is a huge and very expensive project. If the government were to runs this organization, then they would have to hire experts and provide salaries, health care and life insurance to them. Similarly, the government would have to buy expensive gear and invest in the search and rescue training programs. This mission is full of responsibilities. 

For example, if someone might be injured, the government would be responsible for all expenditures or someone were to go missing or die unfortunately, so their family members might be able to claim compensation. As a result, the government might have to pay them. This mission needs lots of money to run because the government thinks this mission should be run by a non profit organization, and that's why they encourage local non profit authorities. In fact the government already maintain emergencies rescue services such as, fire department, forest rangers and police etc; however, there are not enough rescue workers to cover all the places in the country. For instance, there are many people who get stuck in the forest while hiking or climbing, so many rescue workers would be needed for help to them. It might be difficult for the government to send rescue workers to many places. State government encourage non profit organization such as Crag Rats who love and passionate about their work. Their effort of volunteering saves many people's life, and this is the spirit for Crag Rats to continue thier work since old times.