It is better to get injured on some mountains than others. Even which side of the mountain range on which you are injured can determine the quality of response.
The above scenario would probably require a mobile phone and many (including me) do not carry one most of the time.
It would be sad if someone had to spend two chilly nights out w/o pain relief rather than one if the "personal crew" couldn't get the job done.
Most rescue units have liability insurance to protect them when operating under the county, or they have their own unit policies in place. Make sure these are really good friends.
Be sure you have some rock solid medical background to work with. A distorted tib-fib fx can easily distract someone from an aching spine, for example, and shock must be recognized and the source identified.