Why have the AA attend only to find they need to call out SAR, increasing the time it takes to get the patient to safety plus paying for two aircraft and two crew?
Because the AA is a first response vehicle for the Ambulance Sevice, and SAR is a specialist asset which can be kept up your sleeve for appropriate taskings.
Generally speaking the type of civvy taskings SAR usually get involved with will be remote locations or difficult access. Note that these two situations are not necessarily one and the same.
Quite often the AA will be first on scene, and will attend to the patient. IF the access is difficult, or the AA had to land some way off the patient giving a lengthy or difficult extraction by MRT, SAR may be requested to winch.
It's all a case of an appropriate response based on the information available at the time of call. No point calling SAR out for every job when 9 out of 10 of them wouldn't require their specialist skills.
FF