It seems like charging people for the cost of the rescue would solve that problem.
It's not just a theoretical possibility that people will hesitate to call for rescue if they think they'll be charged for it, it's something that has already happened in a lot of cases. When people delay a call for rescue, it complicates search and rescue efforts and may even further endanger rescuers.
A great example of this is injuries that worsen to the point that they require evac by helicopter. Helicopters are an over-used tool in SAR and they are extremely dangerous. It's much safer for us if we can just hike out to you, triage a minor injury, and escort you out on foot or by litter if necessary. If we arrive and you're delirious with pain, exhaustion, or dehydration, those options have just gone off the table.
Another really unfortunate related issue is the number of people that think they have to wait a certain period of time before reporting someone overdue or missing. That's not at all the case, especially where children are involved. We would much rather get out while the potential search area is only a few square miles than get called when it's gone up to 30 or more square miles, even if it means we more frequently get woken up in the middle of the night only to get a "disregard, subject located" notification 20 minutes later.
I have a number of statements available from various SAR groups and other articles on this subject if anyone's interested.
I bought rescue insurance the next day.
However, people that spend much time away from civilization ought to have Global Rescue or something similar. I support government and non profit rescue efforts and funding, but ultimately people participating in relatively risky adventures should bear some responsibility for having rescue insurance — and such insurance is pretty cheap. I’ve seen hikers with $300 hiking poles, $300 hiking boots, expensive GoreTex and yet they scoff at paying for insurance to cover the very expensive costs for rescue. A helicopter can cost $1000+ per hour to operate and people going into the backcountry are assuming a level of risk that they ought to be responsible for insuring.