Three kind choppers used here (Northern CA).

Military - Naval, Coast Guard
They use rescues as part of their training, we often have them
out on our trainings too. This is paid for by taxes either
way. If there was no rescue, they would be doing mock rescues
etc. and spending the same amount of money.

State Highway Patrol
Tax dollars from the state. Somewhat similar to the military,
they would have the choppers in the air anyway, tho unlike the military there could be overtime pay for the pilots.

Private Ambulance
The person who rides, pays.

The military ones are the most capable, but not always available. We most often see Highway Patrol on searches.

I see SAR like the fire department, its part of what I think
should be covered by taxes, but should make use of volunteers
as much as can be done. Most house fires probably can be prevented by forethought, but I don't think the unfortunate
should then get a bill after their house burns. Unless they
are arsonists.

Most search and rescue recovery's are by foot or snowmobile/ATV
anyway. The gas and other gear is paid for by the volunteers and funds from donations.