The publication 'Aircraft Accidents - Guidance for Police, Emergency Services and Airfield Operators' (2008) describes the role of the AAIB but not really to the depth you are looking for.
If you read reports produced by the AAIB it is clear that, sometimes, they try to get close to some of the aspects you refer to. However, from the outside looking in it seems to me that a) they might not have the resources b) they maybe don't have the analytical tools needed to do the sort of investigation you are talking about and c) when they do get interested in organizational factors it is maybe dealt with outside of the report i.e they sit down with the operator, the CAA and some sharp pointy sticks. Which is why sometimes the reports seem to written in code.