That sort of system works in the Alps, but I don't think it translates so well to much of the U.S. or Canada. Note that Switzerland is a very tiny country (compared to the US). Rescue services (and billing, and insurance) can be readily centralized. Most SAR activity is for climbers, hikers, and skiers. Rescues are frequent enough to justify a full time dedicated service.

In contrast, the U.S. and Canada are vast countries with a huge variety of SAR problems. Depending on where you get into trouble (and what sort of trouble you are in) responsibility for rescue may fall upon local police or fire departments, county sheriffs, state police, the Coast Guard, National Park Service, the military, volunteer SAR teams, or other agencies. Missions can involve everything from climbers dangling on cliffs to demetia patients who wander into the woods at the edge of town to commercial fishermen in the Beriing Sea. In some areas SAR missions are very frequent, while in other places they are rather uncommon. Trying to put all of this together into a universal pay-for-rescue service such as in Switzerland would be a logistical and administrative nightmare.

Note that we do have something like the European system in a few areas where rescues are frequent and expensive. On Denali for example, the NPS maintains a professional high altitude rescue team and helicopter during the climbing season. All climbers are required to pay a fee before starting their climb. The fee goes into a fund that helps off set the cost of rescues. That works there because it is a local area of frequent and expensive rescues administered by one agency.