Die Aussage 'ich agiere nicht nur unparteiisch, ...' wäre im deutschen Kontext strenggenommen allerdings auch schon falsch. Denn ich agiere nie unparteiisch, immer allparteilich.
Aber zurück zur Frage: Zugänglichkeit von 'omnipartial(ity)'
Würde man diese Erklärung verstehen?:
Omnipartiality is made up of 'omni' and 'partiality'.
Partiality comes from the Lat. 'pars' resp. 'partialitas' and means 'part' resp. 'participation'.
If one speaks of conflicting parties in the context of mediation, then the parts of the conflict are meant. Or in other words: The people, that are participating in a conflict.
If one speaks of the ability of a mediator, what is meant is his ability to engage with the conflicting parties.
Omni also comes from the Lat. and means something like everyone, all.
Accordingly, when the mediator is spoken of omnipartiality, then nothing else is meant than an attitude of the mediator who is friendly towards all parties to the conflict.