Thought the UK perspective ( oh no not again I hear you cry ) may be of interest.
Item one - our police are not routinely armed, so very unlikely any RTA etc would be attended by an armed officer.
Most of our armed response units went through a phase of trying to deal with injured animals, runaway bulls etc - soon to realise they were simply not trained nor equipped to do it safely.
It has been long extablished practise for police forces in areas prone to deer collisions etc to have a rota of persons prepared to voluntarily attend road scenes and despatch injured deer. These tend to be Gamekeepers, estate staff, zoo staff or keen amateur stalkers etc - plus occassionally Vets.
Much less ' sue 'em all' over here - so this has gone on in a largely informal manner for years. Only recently have liability and insurance issues started in.
All persons holding a FAC have been police vetted and are bound to be of a reasonably responsible nature. In addition, many of the stalkers voluntarily take a two part examination - so are effectively qualified.
All this presumes the use of a firearm. Many times it is not safe to use full -bore rifle. Decscending choice is then a .22 rf ( not legal in all parts ) or a shotgun - both at close range. Thereafter either a knife applied to sever the cervical column or thrust into side of neck and driven forwards to cut arteries. Less common, but effective is a 2 lb- 4lb lump hammer applied resolutely to the base of the ear from behind.
The closer you get, the more risk of injury you run. What many antis or none hunters can never understand is that most stalkers/ responders over here will run the risk of injury to put an animal out of its suffering - we do actually care for these creatures.