Helinut,
With regard to your first paragraph, not quite; police aircraft can do patient transfers, blood/organs etc as well as casevac, but without a HEMS approval none of it can be chargeable, and because the freedoms for landing sites etc excluding the casevac case are often impractical for short notice events, then casevac is all the police end up doing.
The charitable status of our local air ambulance seems fairly well known, I can't really comment about the rest of the country.
Daft tasking? We all suffer from that - as I'm sure you know the average police unit turns down at least three quarters of the the requests that come in. It's just slightly depressing that because of the lack of joined up emergency services, you can have one government funded service available H24 and a charitably funded one available 10 hours each costing about the same, working about 4 hours a day between them, rarely airborne at the same time. Go figure.