I just wonder if Admin have easy access to data about the actions Moderators are taking.
Who knows - it depends upon the capabilities and configuration of the software, as well as the wishes of Admin.
When I've been in such a role in the fairly distant past (in relation to 'commercial' forums), all activities of 'mods' were logged, but goodness knows whether anyone ever looked at the logs. Similarly, when one 'removed' a post, it wasn't deleted but was moved to a private 'forum' that was visible to the entire 'mod'/Admin team - not the least so that 'offending posts' were retained in case they were needed as 'evidence' to justify other actions against repeat offenders in the future - the same might well happen here. There were also private 'sub forums' where all members of the mod/Admin team discussed many/most of the actions they were taking - so the entire team was kept fairly well informed.
The few (non-commercial) forums where I currently (in 2018) have managerial powers are a bit different, in that they don't really have a hierarchical management structure, but the software doesn't seem to include any logging of managerial actions or even retention of deleted messages. There is a list of 'banned members' (including date of banning, but no indication of who banned them), but that seems to be about all.
However, as John has been saying, what we really want is a system with which everyone is 'happy' and which works well, rather than one which 'works badly' but with provision for investigating who is responsible for the 'badness'.
As regards issues of alleged 'bias' or 'double standards' on the part of the management team, one issue might be that of the 'past history' of members involved, since 'persistently serial offenders' are likely to be regarded and treated differently from those with a lesser (or non-existent) history of 'offences'. I would imagine that it could be quite telling to see statistics relating to the number of times each member had had a post removed or edited, had been responsible for a thread being locked, had been given a 'warning', had been locked out of a thread (or temporarily from the whole forum), had had restrictions imposed on their posting (e.g. moderation of posts) etc. etc. There might well be a clear 'record holder'.
Kind Regards, John