My breakdown company came and recovered my vehicle after a hit and run. Their bill was part of the claim

Perhaps it depends upon how bad the acident/damage is - if bits of your car are all over the road, etc, then the scene needs a professional firm, not a jack of all trades.

As HJ said, I seriously doubt if a breakdown operative would want to hang around whilst the scene is firstly made safe, then pick up all the bits and pop them in his van. Not so bad if its just a bump and there no/little debris to clear away.