Helicopter costs can be an issue, depending on the circumstances.

Here in Alaska, we have Air National Guard rescue squadrons available (PJs, Pavehawks, and Hercs). They do civil SAR as part of their training. However, their participation depends on certain parameters. Their first priority is military SAR, which sometimes makes them unavailable for civil SAR. Also, they can't compete with private industry, so they won't accept a medevac mission if a private company is available and willing to do the mission. And they don't do civilian body recovery missions.

The Alaska State Troopers have a couple of A-Star helos for SAR. But they do have a fixed yearly budget for SAR missions. I've never heard of them turning down a real life or death mission for budget considerations, but they might sometimes ask if there is some other way to do the mission. Or ask if the helo is really the most appropriate tool for a particular situation.

What occasionally happens is family and friends often want more assets than are available from the Air National Guard, or the Troopers. Or in some cases, after days of searching with no results the official search is called off, family/friends are not willing to give up. They will sometimes do a go fund me effort to hire a contract helo. I've even heard of instances where there were safety issues with private helos operating in the same air space with "official" aircraft, without proper coordination.